A New Class of Transport Distances Between Measures Jean Dolbeault, Bruno Nazaret, Giuseppe Savaré

A new class of transport distances between measures Jean Dolbeault, Bruno Nazaret, Giuseppe Savaré

To cite this version:

Jean Dolbeault, Bruno Nazaret, Giuseppe Savaré. A new class of transport distances between measures. Calc. Var. Partial Differential Equations, 2009, 34 (2), pp.193-231. hal-00262455

HAL Id: hal-00262455 https://hal.archives-ouvertes.fr/hal-00262455 Submitted on 11 Mar 2008

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A new class of transport distances between measures

Jean Dolbeault Bruno Nazaret Giuseppe Savare´ · ·

March 11, 2008

Abstract We introduce a new class of distances between nonnegative Radon measures in Rd. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein- Wasserstein distances proposed by BENAMOU-BRENIER [7] and provide a wide family in- 1,p terpolating between the Wasserstein and the homogeneous Wγ− -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configu- rations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich- Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given. Keywords Optimal transport Kantorovich-Rubinstein-Wasserstein distance Continuity equation Gradient flows · · ·

1 Introduction

Starting from the contributions by Y. BRENIER,R.MCCANN,W. GANGBO,L.C. EVANS, F. OTTO,C.VILLANI [9,18,25,17,27], the theory of Optimal Transportation has received a lot of attention and many deep applications to various mathematical ﬁelds, such as PDE’s, Calculus of Variations, functional and geometric inequalities, geometry of metric-measure spaces, have been found (we refer here to the monographs [28,16,30,3,31]). Among all possible transportation costs, those inducing the so-called Lp-KANTOROVICH-RUBINSTEIN-

J. Dolbeault, B. Nazaret Ceremade (UMR CNRS no. 7534), UniversitéParis-Dauphine, place de Lattre de Tassigny, 75775 Paris Cédex 16, France. E-mail: {dolbeaul, nazaret}@ceremade.dauphine.fr G. Savaré Università degli studi di Pavia, Department of Mathematics, Via Ferrata 1, 27100, Pavia, Italy. E-mail: [email protected] 2

WASSERSTEIN distances Wp(µ0, µ1), p (1,+∞), between two probability measures µ,ν ∈ ∈ P(Rd) 1 p p Wp(µ0, µ1) := inf y x dΣ : Σ Γ (µ0, µ1) (1.1) Rd Rd | − | ∈ (Z × ) play a distinguished role. Here Γ (µ0, µ1) is the set of all couplings between µ0 and µ1: they are probability measures Σ on Rd Rd whose first and second marginals are respectively d × d d µ0 and µ1, i.e. Σ(B R ) = µ0(B) and Σ(R B) = µ1(B) for all Borel sets B B(R ). It was one of the× most surprising achievement× of [24,25,19,26] that many∈ evolution partial differential equations of the form δ F q 2 d ∂t ρ + ∇ ρ ξ − ξ = 0, ξ = ∇ in R (0,+∞), (1.2) · − δρ × can be, at least formally, interpreted as gradient flows of suitable integral functionals F with respect to Wp (see also the general approach developed in [30,3,31]). In (1.2) δ F /δρ is the Euler first variation of F , q := p/(p 1) is the Hölder’s conjugate exponent of p, − and t ρt (a time dependent solution of (1.2)) can be interpreted as a flow of probability 7→ d d d measures µt = ρt L with density ρt with respect to the Lebesgue measure L in R . Besides showing deep relations with entropy estimates and functional inequalities [27], this point of view provides a powerful variational method to prove existence of solutions to (1.2), by the so-called Minimizing movement scheme [19,13,3]: given a time step τ > 0 and d d an initial datum µ0 = ρ0L , the solution µt = ρt L at time t nτ can be approximated by n ≈ the discrete solution µτ obtained by a recursive minimization of the functional 1 µ W p(µ, µk) + F (µ), k = 0,1, (1.3) τ p 1 p τ 7→ p − ··· The link between the Wasserstein distance and equations exhibiting the characteristic struc- ture of (1.2) (in particular the presence of the diffusion coefficient ρ, the fact that ξ is a gradient vector field, and the presence of the q-duality map ξ ξ q 2 ξ ), is well explained 7→ | | − by the dynamic characterization of Wp introduced by BENAMOU-BRENIER [7]: it relies in the minimization of the “action” integral functional

1 p µ µ ρ p Wp ( 0, 1) = inf t (x) vt (x) dxdt : 0 Rd | | (1.4) nZ Z ∂ ρ + ∇ (ρ v ) = 0 in Rd (0,1), µ = ρ L d, µ = ρ L d . t t t t 0 t 0 1 t 1 · × | = | = o Towards more general cost functionals. If one is interested to study the more general class of diffusion equations δ F q 2 d ∂t ρ + ∇ h(ρ) ξ − ξ = 0, ξ = ∇ in R (0,+∞), (1.5) · − δρ × obtained from (1.2) replacing the mobility coefﬁcient ρ by an increasing nonlinear function h(ρ), h : [0,+∞) [0,+∞) whose typical examples are the functions h(ρ) = ρα , α 0, it is then natural to→ investigate the properties of the “distance” ≥

1 p µ µ ρ p Wp ( 0, 1) = inf h t (x) vt (x) dxdt : 0 Rd | | (1.6) nZ Z ∂ ρ + ∇ (h(ρ )v ) = 0 inRd (0,1), µ = ρ L d, µ = ρ L d . e t t t t 0 t 0 1 t 1 · × | = | = o 3

In the limiting case α = 0, h(ρ) 1, one can easily recognize that (1.6) provides an equiv- ≡ 1,p d alent description of the homogeneous (dual) W˙ − (R ) Sobolev (pseudo)-distance

µ µ ζ µ µ ζ 1 Rd ζ q 0 1 W˙ 1,p(Rd) := sup d( 0 1) : Cc ( ), D dx 1 . (1.7) k − k − Rd − ∈ Rd | | ≤ nZ Z o Thus the distances deﬁned by (1.6) for 0 α 1 (we shall see that this is the natural range for the parameter α) can be considered≤ as≤ a natural “interpolating” family between the Wasserstein and the (dual) Sobolev ones. Notice that if one wants to keep the usual transport interpretation given by a “dynamic cost” to be minimized along the solution of the continuity equation, one can simply introduce ρ 1 ρ the velocity vector ﬁeld v˜t := t− h( t )vt and minimize the cost

1 ρ p 1 p − ρ f (ρ) v˜t dxdt, where f (ρ) := . (1.8) 0 Rd | | h(ρ) Z Z p Therefore, in this model the usual p-energy Rd ρt v˜t dx of the moving masses ρt with velocity v˜ results locally modiﬁed by a factor f ρ | depending| on the local density of the t R ( t) mass occupied at the time t. Different non-local models have been considered in [8,4]. In the present paper we try to present a systematic study of these families of intermediate distances, in view of possible applications, e.g., to the study of evolution equations like (1.5), the Minimizing movement approach (1.3), and functional inequalities.

Examples: PDE’s as gradient ﬂows. Let us show a few examples evolution equations which can be formally interpreted as gradient ﬂows of suitable integral functionals in this setting: the scalar conservation law

α ∂tρ ∇ ρ ∇V = 0 corresponds to the linear functional F (ρ) := V(x)ρ dx, Rd − · Z for some smooth potential V : Rd R and p = 2. Choosing for m > 0 → m+1 α m p = 2, F (ρ) = cα ρ − dx, cα := , ,m ,m (m + 1 α)(m α) Z − − one gets the porous media/fast diffusion equation

m α m α m ∂t ρ ∇ ρ ∇ρ − = ∂t ρ ∆ρ = 0, (1.9) − m α · − − 1 ρ2 α and in particular the heat equation for the entropy functional (2 α)(1 α) − dx. Choosing − − R m+2q 3 α m(q 1)q F (ρ) = cα ρ q −1 − dx, cα := − , ,m,q − ,m,q (m + 2q 3 α)(m + q 2 α) Z − − − − one obtains the doubly nonlinear equation

m 1 q 2 ∂t ρ m∇ (ρ − ∇ρ − ∇ρ) = 0 (1.10) − · | | and in particular the evolution equation for the q-Laplacian when m = 1. The Dirichlet integral for p = 2

1 2 α F (ρ) = ∇ρ dx yields ∂t ρ + ∇ ρ ∇∆ρ = 0, (1.11) 2 | | · Z a thin-ﬁlm like equation. 4

The measure-theoretic point of view: Wasserstein distance. We present now the main points of our approach (see also, in a different context, [10]). First of all, even if the language of densities and vector ﬁelds (as ρ and v,v˜ in (1.4) or (1.6)) is simpler and suggests interesting interpretations, the natural framework for considering the variational problems (1.4) and (1.6) is provided by time dependent families of Radon measures in Rd. Following this d point of view, one can replace ρt by a continuous curve t [0,1] µt (µt = ρt L in the M+ Rd ∈ 7→ Rd absolutely continuous case) in the space loc( ) of nonnegative Radon measures in 0 Rd endowed with the usual weak∗ topology induced by the duality with functions in Cc ( ). The (Borel) vector ﬁeld vt in (1.4) induces a time dependent family of vector measures ν t := µt vt µt . In terms of the couple (µ,ν) the continuity equation (1.4) reads ≪ d ∂t µt + ∇ ν t = 0 in the sense of distributions in D′(R (0,1)), (1.12) · × and it is now a linear equation. Since vt = dν t /dµt is the density of ν t w.r.t. µt , the action functional which has to be minimized in (1.4) can be written as 1 dν p Ep,1(µ,ν) = Φp,1(µt ,ν t )dt, Φp,1(µ,ν) := dµ. (1.13) Rd dµ Z0 Z

d µ ρ d Notice that in the case of absolutely continuous measures with respect to L , i.e. = L d and ν = wL , the functional Φp,1 can also be expressed as p d w Φp,1(µ,ν) := φp,1(ρ,w)dL (x), φp,1(ρ,w) := ρ . (1.14) Rd ρ Z

CE µ ν Denoting by (0,1) the class of measure-valued distributional solutions , of the continuity equation (1.12), we end up with the equivalent characterization of the Kantorovich- Rubinstein-Wasserstein distance

W p(µ , µ ) := inf E (µ,ν) : (µ,ν) CE(0,1), µ = µ , µ = µ . (1.15) p 0 1 p,1 t 0 0 t 1 1 ∈ | = | = n o d Structural properties and convexity issues. The density function φ = φp,1 : (0,+∞) R Rd appearing in (1.14) exhibits some crucial features × → 1. w φ( ,w) is symmetric, positive (when w = 0), and p-homogeneous with respect 7→ · 6 to the vector variable w: this ensures that Wp is symmetric and satisfies the triangular inequality. d 2. φ is jointly convex in (0,+∞) R : this ensures that the functional Φp 1 (and therefore × , also E ) defined in (1.13) is lower semicontinuous with respect to the weak∗ convergence of Radon measures. It is then possible to show that the infimum in (1.15) is attained, as soon as it is finite (i.e. when there exists at least one curve (µ,ν) CE(0,1) with finite ∈ energy E (µ,ν) joining µ0 to µ1); in particular Wp(µ0, µ1) = 0 yields µ0 = µ1. Moreover, the distance map (µ0, µ1) Wp(µ0, µ1) is lower semicontinuous with respect to the 7→ weak∗ convergence, a crucial property in many variational problems involving Wp, as (1.3). 3. φ is jointly positively 1-homogeneous: this a distinguished feature of the Wasserstein case, which shows that the functional Φp,1 depends only on µ,ν and not on the Lebesgue measure L d , even if it can be represented as in (1.14). In other words, suppose that µ = ργ˜ and ν = w˜ γ, where γ is another reference (Radon, nonnegative) measure in Rd. Then

Φp,1(µ,ν) = φp,1(ρ˜,w˜ )dγ. (1.16) Rd Z 5

As we will show in this paper, the 1-homogeneity assumption yields also two “quantita- tive” properties: if µ0 is a probability measure, then any solution (µ,ν) of the continuity d equation (1.12) with finite energy E (µ,ν) < +∞ still preserves the mass µt (R ) 1 for every time t 0 (and it is therefore equivalent to assume this condition in the def-≡ ≥ inition of CE(0,1), see e.g. [3, Chap. 8]). Moreover, if the p-moment of µ0 mp(µ0) := p Rd x dµ0(x) is finite, then Wp(µ0, µ1) < +∞ if and only if mp(µ1) < +∞. | | R Main definitions. Starting from the above remarks, it is then natural to consider the more general case when the density functional φ : (0,+∞) Rd [0,+∞) still satisfies 1. (p- homogeneity w.r.t. w) and 2. (convexity), but not 3. (1-homogeneity).× → Due to this last choice, the associated integral functional Φ is no more independent of a reference measure γ and it seems therefore too restrictive to consider only the case of the Lebesgue measure γ = L d. In the present paper we will thus introduce a further nonnegative reference Radon mea- γ M+ Rd φ ∞ Rd ∞ sure loc( ) and a general convex functional : (0,+ ) [0,+ ) which is p-homogeneous∈ w.r.t. its second (vector) variable and non degenerate× → (i.e. φ(ρ,w) > 0 if w = 0). Particularly interesting examples of density functionals φ, corresponding to (1.6), are6 given by w p φ(ρ,w) := h(ρ) , (1.17) h(ρ)

where h : (0,+∞) (0,+∞) is an increasing and concave function; the concavity of h is a necessary and→ sufﬁcient condition for the convexity of φ in (1.17) (see [29] and §3). Choosing h(ρ) := ρα , α (0,1), one obtains ∈ p α w θ p p φp α (ρ,w) := ρ = ρ − w , θ :=(1 α) p + α (1, p), (1.18) , ρα | | − ∈

which is jointly θ-homogeneous in (ρ,w). In the case, e.g., when α < 1 in (1.18) or more generally limρ ∞ h(ρ)/ρ = 0, the recession function of φ satisﬁes ↑

∞ 1 φ (ρ,w) = lim λ − φ(λρ,λw) = +∞ if ρ,w = 0, (1.19) λ +∞ 6 ↑ so that the associated integral functional reads as

Φ(µ,ν γ) := φ(ρ,w)dγ µ = ργ + µ⊥, ν = wγ γ, (1.20) Rd | Z ≪ extended to +∞ when ν is not absolutely continuous with respect to γ or supp(µ) supp(γ). Notice that only the density ρ of the γ-absolutely continuous part of µ enters in6⊂ the functional, but the functional could be ﬁnite even if µ has a singular part µ⊥. This choice is crucial in order to obtain a lower semicontinuous functional w.r.t. weak∗ convergence of measures. The associated (φ,γ)-Wasserstein distance is therefore

p W (µ , µ ) := inf Eφ γ (µ,ν) : (µ,ν) CE(0,1), µ = µ , µ = µ , (1.21) φ,γ 0 1 , t 0 0 t 1 1 ∈ | = | = n o where the energy Eφ γ of a curve (µ,ν) CE(0,1) is , ∈ 1 Eφ,γ (µ,ν) := Φ(µt ,ν t γ)dt. (1.22) 0 | Z 6

The most important case associated to the functional (1.18) deserves the distinguished notation

Wp α;γ ( , ) := Wφ α γ ( , ). (1.23) , · · p, , · · p 1,p The limiting case α = θ = 1 corresponds to the L -Wasserstein distance, the Sobolev W˙ γ− corresponds to α = 0, θ = p. The choice of γ allows for a great ﬂexibility: besides the Lebesgue measure in Rd, we quote γ L d Ω Rd Ω¯ – := Ω , being an open subset of . The measures are then supported in and, with the| choice (1.17) and v = w/h(ρ), (1.12) is a weak formulation of the continuity equation (n∂Ω being the exterior unit normal to ∂Ω)

∂t ρt + ∇ h(ρt )vt = 0 in Ω (0,1), vt n∂Ω = 0 on ∂Ω. (1.24) · × · This choice is useful for studying equations (1.9) (see [11]), (1.10), (1.11) in bounded domains with Neumann boundary conditions. V d 1 d – γ := e− L for some C potential V : R R. With the choice (1.17) and v = w/h(ρ) (1.12) is a weak formulation of the equation→

d ∂t ρt + ∇ h(ρt )vt h(ρt )∇V vt = 0 in R (0,1). (1.25) · − · × ρ ρα F µ 1 ρ2 α γ When h( ) = , p = 2, the gradient ﬂow of ( ) := (2 α)(1 α) Rd − d is the − − Kolmogorov-Fokker-Planck equations [15] R

∂t µ ∆ µ ∇ (µ∇V) = 0, ∂t ρ ∆ρ + ∇V ∇ρ = 0, − − · − · which in the Wasserstein framework is generated by the logarithmic entropy ([19,3,5]). γ H k M Rd – := M, being a smooth k-dimensional manifold embedded in with the Rie- mannian| metric induced by the Euclidean distance; H k denotes the k-dimensional Haus- dorff measure. (1.12) is a weak formulation of

∂t ρt + divM h(ρ)vt = 0 on M (0,1). (1.26) × Thanks to Nash embedding theorems [22,23], the study of the continuity equation and of the weighted Wasserstein distances on arbitrary Riemannian manifolds can be reduced to this case, which could be therefore applied to study equations (1.9), (1.10), (1.11) on Riemannian manifolds.

Main results. Let us now summarize some of the main properties of Wp,α;γ ( , ) we will prove in the last section of the present paper. In order to deal with distances· (instead· of pseudo-distances, possibly assuming the value +∞), for a nonnegative Radon measure σ we will denote by Mp,α;γ [σ] the set of all measures µ with Wp,α;γ (µ,σ) < +∞ endowed with the Wp,α;γ -distance.

1. Mp,α;γ [σ] is a complete metric space (Theorem 5.7). 2. Wp,α;γ induces a stronger convergence than the usual weak∗ one (Theorem 5.5). 3. Bounded sets in Mp,α;γ [σ] are weakly∗ relatively compact (Theorem 5.5). 4. The map (µ0, µ1) Wp,α;γ (µ0, µ1) is weakly∗ lower semicontinuous (Theorem 5.6), convex (Theorem 5.11),7→ and subadditive (Theorem 5.12). It enjoys some useful monotonicity properties with respect to γ (Proposition 5.14) and to convolution (Theorem 5.15). 7

5. The inﬁmum in (1.15) is attained, Mp,α;γ [σ] is a geodesic space (Theorem 5.4), and constant speed geodesics connecting two measures µ0, µ1 are unique (Theorem 5.11). 6. If

p/(θ 1) p q x − − dγ(x) < +∞ θ =(1 α)p + α, = , (1.27) x 1 | | − θ 1 1 α Z| |≥ − − d d and σ P(R ), then Mp α;γ [σ] P(R ) (Theorem 5.8). If moreover γ satisﬁes stronger ∈ , ⊂ summability assumptions, then the distances Wp,α;γ provide a control of various mo- 1,p ments of the measures (Theorem 5.9). Comparison results with Wp and W˙ − are also discussed in §5.4. 7. Absolutely continuous curves w.r.t. Wp,α;γ can be characterized in completely analogous ways as in the Wasserstein case (§5.3). 8. In the case γ = L d the functional

1 2 α d d Ψα (µ γ) := ρ − dx µ = ρL L , (1.28) | (2 α)(1 α) Rd ≪ − − Z Rd is geodesically convex w.r.t. the distance W2,α;L d and the heat equation in is its gradient ﬂow, as formally suggested by (1.9) (§5.5: we prove this property in the case d α > 1 2/d, when P(R ) is complete w.r.t. W α L d .) − 2, ;

Plan of the paper. Section 2 recalls some basic notation and preliminary facts about weak∗ convergence and integral functionals of Radon measures; 2.3 recalls a simple duality result in convex analysis, which plays a crucial role in the analysis of the integrand φ(ρ,w). The third section is devoted to the class of admissible action integral functionals Φ like (1.20) and their density φ. Starting from a few basic structural assumptions on φ we deduce its main properties and we present some important examples in Section 3.2. The corresponding properties of Φ (in particular, lower semicontinuity and relaxation with respect to weak∗ convergence, monotonicity, etc) are considered in Section 3.3. Section 4 is devoted to the study of measure-valued solutions of the continuity equation (1.12). It starts with some preliminary basic results, which extend the theory presented in [3] to the case of general Radon measures: this extension is motivated by the fact that the class of probability measures (and therefore with finite mass) is too restrictive to study the d distances Wp,α;γ , in particular when γ(R ) = +∞ as in the case of the Lebesgue measure. P Rd We shall see (Remark 5.27) that ( ) with the distance Wp,α;L d is not complete if d > p/(θ 1) = q/(1 α). We consider in Section 4.2 the class of solutions of (1.12) with finite − − energy Eφ,γ (1.22), deriving all basic estimates to control their mass and momentum. As we briefly showed, Section 5 contains all main results of the paper concerning the modified Wasserstein distances.

2 Notation and preliminaries

Here is a list of the main notation used throughout the paper: h BR The open ball (in some R ) of radius R centered at 0 h h h B(R ) (resp. Bc(R )) Borel subsets of R (resp. with compact closure) P(Rh) Borel probability measures in Rh M+ Rh M+ Rh Rh ( ) (resp. loc( )) Finite (resp. Radon), nonnegative Borel measures on P(Rh) Borel probability measures in Rh 8

M(Rh;Rm) Rm-valued Borel measures with ﬁnite variation h m m Mloc(R ;R ) R -valued Radon measures h m µ Total variation of µ Mloc(R ;R ), see (2.2) k 0 kRh ∈ Cb ( ) Continuous and bounded real functions p + h mp(µ) p-moment Rd x dµ of µ M (R ) ψ∞ Recession function| | of ψ, see∈ (2.4) R Ψ(µ γ), Φ(µ,ν γ) Integral functionals on measures, see 2.2 and 3.3 | | µ(ζ), µ,ζ , µ ,ζ the integrals Rd ζ dµ, Rd ζ dµ h i h i · CE(0,T),CEφ γ (0,T), Classes of measure-valued solutions of the continuity , R R CE(0,T; µ0 µ1) equation, see Def. 4.2 and Sec. 4.2. →

2.1 Measures and weak convergence

We recall some basic notation and properties of weak convergence of (vector) radon mea- h m m sures (see e.g. [2]). A Radon vector measure in Mloc(R ;R ) is a R -valued map µ : h m h Bc(R ) R defined on the Borel sets of R with compact closure. We identify µ h→ m 1 2 m h ∈ Mloc(R ;R ) with a vector (µ ,µ , ,µ ) of m measures in Mloc(R ): its integral with a continuous vector valued function with··· compact support ζ C0(Rh;Rm) is given by ∈ c m µ ,ζ := ζ dµ = ∑ ζ i(x)dµ i(x). (2.1) Rh Rh h i Z · i=1 Z M Rh Rm 0 Rh Rm It is well known that loc( ; ) can be identified with the dual of Cc ( ; ) by the above duality pairing and it is therefore endowed with the corresponding of weak∗ topology. If is a norm in Rd with dual (in particular the euclidean norm ) for every open subsetk · k A Rh we have k · k∗ |·| ⊂ µ (A) = sup ζ dµ : supp(ζ ) A, ζ (x) 1 x Rh . (2.2) k k Rh · ⊂ k k∗ ≤ ∀ ∈ nZ o µ M+ Rh µ is in fact a Radon positive measure in loc( ) and admits the polar decomposition µk =k w µ where the Borel vector field w belongs to L1 ( µ ;Rm). We thus have k k loc k k

µ,ζ = ζ dµ = ζ wd µ . (2.3) h i Rh · Rh · k k Z Z µ M Rh Rm µ ∞ If ( k)k N is a sequence in loc( ; ) with supn (BR) < + for every open ball ∈ k k h m BR, then it is possible to extract a subsequence µ weakly convergent to µ M(R ;R ), kn ∗ ∈ whose total variation µ weakly converges to λ M+(Rh) with µ λ. k kn k ∗ ∈ k k ≤

2.2 Convex functionals deﬁned on Radon measures

Let ψ : Rm [0,+∞] be a convex and lower semicontinuous function with ψ(0) = 0, whose proper domain→ D(ψ) := x Rm : ψ(x) < +∞ has non empty interior. Its recession function ∞ { ∈ } (see e.g. [2]) ψ : Rm [0,+∞] is deﬁned as → ∞ ψ(ry) ψ(ry) ψ (y) := lim = sup . (2.4) r +∞ r r → r>0 9

ψ∞ is still convex, lower semicontinuous, and positively 1-homogeneous, so that its proper domain D(ψ∞) is a convex cone always containing 0. We say that

∞ ∞ ψ has a superlinear growth if ψ (y) = ∞ for every y = 0: D(ψ ) = 0 , ∞ 6 { } (2.5) ψ has a sublinear growth if ψ (y) 0 for every y Rm. ≡ ∈ + h h m Let now γ M (R ) and µ Mloc(R ;R ) with supp(µ) supp(γ); the Lebesgue decom- ∈ loc ∈ ⊂ position of µ w.r.t. γ reads µ = ϑϑγ +µ ⊥, where ϑ = dµ /dγ. We can introduce a nonnegative + h Radon measure σ M (R ) such that µ = ϑ ⊥σ σ, e.g. σ = µ and we set ∈ loc ⊥ ≪ | ⊥|

a ∞ ∞ Ψ (µ γ) := ψ(ϑ (x))dγ(x), Ψ (µ γ) := ψ (ϑ ⊥(y))dσ(y), (2.6) | Rh | Rh Z Z and ﬁnally

∞ ∞ Ψ(µ γ) := Ψ a(µ γ) +Ψ (µ γ); Ψ (µ γ) = +∞ if supp(µ) supp(γ). (2.7) | | | | 6⊂ Since ψ∞ is 1-homogeneous, the definition of Ψ ∞ depends on γ only through its support and it is independent of the particular choice of σ in (2.6). When ψ has a superlinear growth then the functional Ψ is finite iff µ γ and Ψ a(µ γ) is finite; in this case Ψ(µ γ) = Ψ a(µ γ). ≪ | | | Theorem 2.1 (L.s.c. and relaxation of integral functionals of measures [1,2]) Let us con- γ M+ Rh µ M Rh Rm γ M+ Rh sider two sequences n loc( ), n loc( ; ) weakly∗ converging to loc( ) h m ∈ ∈ ∈ and µ Mloc(R ;R ) respectively. We have ∈ Ψ µ γ Ψ µ γ liminf ( n n) ( ). (2.8) n +∞ | ≥ | ↑ µ γ Ψ µ γ ∞ µ ϑ γ γ Let conversely , be such that ( ) < + . Then there exists a sequence n = n µ | ≪ weakly∗ converging to µ such that

Ψ a µ γ ψ ϑ γ Ψ µ γ lim ( n ) = lim ( n(x))d (x) = ( ). (2.9) n +∞ | n +∞ Rh | ↑ ↑ Z

Theorem 2.2 (Montonicity w.r.t. γ) If γ1 γ2 then ≤

Ψ(µ γ2) Ψ(µ γ1). (2.10) | ≤ | Proof Thanks to Theorem 2.1, it is sufﬁcient to prove the above inequality for µ γ1. Since i i 2 1 ≪ γ1 = θγ2, with density θ 1 γ2-a.e., we have µ = ϑ γ with ϑ = θϑϑ , and therefore ≤ 1 ψ(ϑ 1)dγ1 = ψ(θ − ϑ 2)θ dγ2 ψ(ϑ 2)dγ2, (2.11) Rd Rd Rd Z Z ≥ Z where we used the property θψ(θ 1x) ψ(x) for θ 1, being ψ(0) = 0. − ≥ ≤ ⊓⊔ Theorem 2.3 (Monotonicity with respect to convolution) If k C∞(Rd) is a convolution ∈ c kernel satisfying k(x) 0, Rd k(x)dx = 1, then ≥ R Ψ(µ k γ k) Ψ(µ γ). (2.12) ∗ | ∗ ≤ | The proof follows the same argument of [3, Lemma 8.1.10], by observing that the map (x,y) xψ(y/x) is convex and positively 1-homogeneous in (0,+∞) Rd. 7→ × 10

2.3 A duality result in convex analysis

Let X,Y be Banach spaces and let A be an open convex subset of X. We consider a convex (and a fortiori continuous) function φ : A Y R and its partial Legendre transform × →

φ˜(x,y∗) := sup y∗,y φ(x,y) ( ∞,+∞], x A, y∗ Y ∗. (2.13) y Y h i − ∈ − ∀ ∈ ∈ ∈ The following duality result is well known in the framework of minimax problems [29].

Theorem 2.4 φ˜ is a l.s.c. function and there exists a convex set Y Y such that o∗ ⊂ ∗

φ˜(x,y∗) < +∞ y∗ Y ∗, (2.14) ⇔ ∈ o so that φ˜( ,y ) +∞ for every y Y Y and φ admits the dual representation formula · ∗ ≡ ∗ ∈ ∗ \ o∗

φ(x,y) = sup y,y∗ φ˜(x,y∗) x A, y Y. (2.15) y Y h i − ∀ ∈ ∈ ∗∈ o∗ For every y Y we have ∗ ∈ o∗

the map x φ˜(x,y∗) is concave (and continuous) in A. (2.16) 7→ Conversely, a function φ : A Y R is convex if it admits the dual representation (2.15) for a function φ˜ satisfying (2.16)×. →

Proof Let us ﬁrst show that (2.16) holds. For a ﬁxed y Y , x0,x1 > 0, θ [0,1], and ∗ ∈ ∗ ∈ arbitrary yi Y, we get ∈

φ˜((1 ϑ)x0 + ϑx1,y∗) y∗,(1 ϑ)y0 + ϑy1 φ((1 ϑ)x0 + ϑx1,(1 ϑ)y0 + ϑy1) − ≥h − i − − − (1 ϑ) y∗,y0 φ(x0,y0) + ϑ y∗,y1 φ(x1,y1) . ≥ − h i − h i − Taking the supremum with respect to y0,y1 we eventually get

φ˜((1 ϑ)x0 + ϑx1,y∗) (1 ϑ)φ˜(x0,y∗) + ϑφ˜(x1,y∗) (2.17) − ≥ − and we conclude that φ˜( ,y ) is concave. In particular, if it takes the value +∞ at some point · ∗ it should be identically +∞ so that (2.14) holds. The converse implication is even easier, since (2.15) exhibits φ as a supremum of continuous and convex functions (jointly in x A,y Y). ∈ ∈ ⊓⊔

3 Action functionals

The aim of this section is to study some property of integral functionals of the type

a + d d d Φ (µ,ν γ) := φ(ρ,w)dγ, µ = ργ M (R ), ν = wγ Mloc(R ;R ) (3.1) Rd loc | Z ∈ ∈ and their relaxation, when φ satisﬁes suitable convexity and homogeneity properties. 11

3.1 Action density functions

Let us therefore consider a nonnegative density function φ : (0,+∞) Rd [0,+∞) and an × → exponent p (1,+∞) satisfying the following assumptions ∈ φ is convex and (a fortiori) continuous, (3.2a)

w φ( ,w) is homogeneous of degree p, i.e. 7→ · (3.2b) φ(ρ,λw) = λ pφ(ρ,w) ρ > 0, λ R, w Rd, | | ∀ ∈ ∈ d ρ0 > 0: φ(ρ0, ) is non degenerate, i.e. φ(ρ0,w) > 0 w R 0 . (3.2c) ∃ · ∀ ∈ \ { } Let q = p/(p 1) (1,+∞) be the usual conjugate exponent of p. We denote by φ˜ : − ∈ (0,+∞) Rd ( ∞,+∞] the partial Legendre transform × → − 1 1 φ˜(ρ,z) := sup z w φ(ρ,w) ρ > 0,z Rd. (3.2d) q w Rd · − p ∀ ∈ ∈ We collect some useful properties of such functions in the following result.

Theorem 3.1 Let φ : (0,+∞) Rd Rd satisfy (3.2a,b,c). Then × → 1. For every ρ > 0 the function w φ(ρ,w)1/p is a norm of Rd whose dual norm is given 7→ by z φ˜(ρ,z)1/q, i.e. 7→ w z w z φ˜(ρ,z)1/q = sup , φ(ρ,w)1/p = sup . (3.3) · 1/p · 1/q w=0 φ(ρ,w) z=0 φ˜(ρ,z) 6 6 In particular φ˜( ,z) is q-homogeneous with respect to z. 2. The marginal conjugate· function φ˜ takes its values in [0,+∞) and for every z Rd ∈ the map ρ φ˜(ρ,z) is concave and non decreasing in (0,+∞). (3.4) 7→ In particular, for every w Rd ∈ the map ρ φ(ρ,w) is convex and non increasing in (0,+∞). (3.5) 7→ 3. There exist constants a,b 0 such that ≥ q 1 p p d φ˜(ρ,z) a + bρ z , φ(ρ,z) a + bρ − w ρ > 0, z,w R . (3.6) ≤ | | ≥ | | ∀ ∈

4. For every closed interval [ρ0,ρ1] (0,+∞) there exists a constant C = Cρ ρ > 0 such ⊂ 0, 1 that for every ρ [ρ0,ρ1] ∈ 1 p p 1 q q d C− w φ(ρ,w) C w , C− z φ˜(ρ,z) C z w,z R . (3.7) | | ≤ ≤ | | | | ≤ ≤ | | ∀ ∈ Equivalently, a function φ satisﬁes (3.2a,b,c) if and only if it admits the dual representation formula 1 1 φ(ρ,w) = sup w z φ˜(ρ,z) ρ > 0,w Rd , (3.8) p z Rd · − q ∀ ∈ ∈ where φ˜ : (0,+∞) Rd (0,+∞) is a nonnegative function which is convex and q-homogeneous w.r.t. z and concave× with→ respect to ρ. 12

Proof Let us ﬁrst assume that φ satisﬁes (3.2a,b,c). The function w φ(ρ,w)1/p is 1- homogeneous and its sublevels are convex, i.e. it is the gauge function7→ of a (symmetric) convex set and therefore it is a (semi)-norm. The concavity of φ˜ follows from Theorem 2.4; taking w = 0 in (3.2d), we easily get that φ˜ is nonnegative; (3.2c) yields, for a suitable constant c0 > 0,

p d q d φ(ρ0,w) c0 w w R , so that φ˜(ρ0,z) c0 z < +∞ z R . (3.9) ≥ | | ∀ ∈ ≤ | | ∀ ∈ Still applying Theorem 2.4, we obtain that ρ φ˜(ρ,z) is ﬁnite, strictly positive and nonde- 7→ creasing in the interval (0,+∞). Since φ˜(ρ,0) = 0 we easily get

q d φ˜(ρ,z) φ˜(ρ0,z) c0 z z R , ρ (0,ρ0); (3.10) ≤ ≤ | | ∀ ∈ ∈ ρ φ˜ ρ φ˜ ρ c0 ρ q Rd ρ ρ ∞ ( ,z) ρ ( 0,z) ρ z z , ( 0,+ ). (3.11) ≤ 0 ≤ 0 | | ∀ ∈ ∈ Combining the last two bounds we get (3.6). (3.7) follows by homogeneity and by the fact that the continuous map φ has a maximum and a strictly positive minimum on the compact d set [ρ0,ρ1] w R : w = 1 . The ﬁnal× { assertion∈ concerning| | } (3.8) still follows by Theorem 2.4. ⊓⊔

3.2 Examples

Example 3.2 Our main example is provided by the function

2 w α 2 φ2 α (ρ,w) = | | , φ˜2 α (ρ,z) := ρ z , 0 α 1. (3.12) , ρα , | | ≤ ≤

Observe that φ2 α is positively θ-homogeneous, θ := 2 α, i.e. , − θ d φ2 α (λρ,λw) = λ φ(ρ,w) λ,ρ > 0, w R . (3.13) , ∀ ∈ It can be considered as a family of interpolating densities between the case α = 0, when

2 φ2 0(ρ,w) := w , (3.14) , | | and α = 1, corresponding to the 1-homogeneous functional

2 φ ρ w 2,1( ,w) := | ρ| . (3.15)

Example 3.3 More generally, we introduce a concave function h : (0,+∞) (0,+∞), which is a fortiori continuous and nondecreasing, and we consider the density function→

w 2 φ(ρ,w) := | | , φ˜(ρ,z) := h(ρ) w 2. (3.16) h(ρ) | |

If h is of class C2, we can express the concavity condition in terms of the function g(ρ) := 1/h(ρ) as 2 h is concave g′′(ρ)g(ρ) 2 g′(ρ) ρ > 0, (3.17) ⇔ ≥ ∀ which is related to a condition introduced in [6, Section 2.2, (2.12c)] to study entropy functionals. 13

Example 3.4 We consider matrix-valued functions H,G : (0,+∞) Md d such that → × 1 H(ρ),G(ρ) are symmetric and positive definite, H(ρ) = G− (ρ) ρ > 0. (3.18) ∀ They induce the action density φ : (0,+∞) Rd [0,+∞) defined as × → 1 φ(ρ,w) := G(ρ)w,w = H− (ρ)w,w . (3.19) h i Taking into account Theorem 3.1, φ satisfies conditions (3.2) if and only if the maps ρ H(ρ)w,w are concave in (0,+∞) w Rd. (3.20) 7→ h i ∀ ∈ Equivalently,

H((1 ϑ)ρ0 + ϑρ1) (1 ϑ)H(ρ0) + ϑH(ρ1) as quadratic forms. (3.21) − ≥ − When G is of class C2 this is also equivalent to ask that

G′′(ρ) 2G′(ρ)H(ρ)G′(ρ) ρ > 0, (3.22) ≥ ∀ 1 in the sense of the associated quadratic forms. In fact, differentiating H = G− with respect to ρ we get H′ = HG′ H, H′′ = HG′′ H + 2HG′ HG′ H, − − so that d2 H(ρ)w,w = G′′w˜ ,w˜ + 2 G′HG′w˜ ,w˜ where w˜ := Hw; d2ρ h i −

we eventually recall that H(ρ) is invertible for every ρ > 0. Example 3.5 Let be any norm in Rd with dual norm , and let h : (0,+∞) (0,+∞) be a concave (continuous,k·k nondecreasing) function as ink·k Exa∗mple 3.3. We can thus→ consider w p φ(ρ,w) := h(ρ) , φ˜(ρ,z) := h(ρ) z q. (3.23) h(ρ) k k∗

See [20,21] for a in-depth study of this class of functions.

Example 3.6 ((α-θ)-homogeneous functionals) In the particular case h(ρ) := ρα the functional φ of the previous example is jointly positively θ-homogeneous, with θ := α +(1 − α)p. This is in fact the most general example of θ-homogeneous functional, since if φ is θ-positively homogeneous, 1 θ p, then ≤ ≤ θ θ θ p α α p p φ(ρ,w) = ρ φ(1,w/ρ) = ρ − φ(1,w) = ρ w/ρ , α = − , (3.24) k k p 1 − where w := φ(1,w)1/p is a norm in Rd by Theorem 3.1. The dual marginal density φ˜ in this casek takesk the form α φ˜(ρ,z) = ρ z q ρ > 0, z Rd, (3.25) k k∗ ∀ ∈ and it is q + α-homogeneous. Notice that α and θ are related by θ α + = 1. (3.26) p q In the particular case when = = is the Euclidean norm, we set as in (3.16) k · k k · k∗ |·| p α w α q φp α (ρ,w) := ρ , φ˜q α (ρ,z) := ρ z , 0 α 1. (3.27) , ρα , | | ≤ ≤

3.3 The action functional on measures

Lower semicontinuity envelope and recession function. Thanks to the monotonicity property (3.5), we can extend φ also for ρ = 0 by setting for every w Rd ∈ φ(0,0) = 0, φ(0,w) = sup φ(ρ,w) = limφ(ρ,w); in particular (3.28) ρ ρ>0 0 φ(0,w) > 0 if w = 0. ↓ ( 6 When ρ < 0 we simply set φ(ρ,w) = +∞, observing that this extension is lower semicontinuous in R Rd . It is not difﬁcult to check that φ˜(0, ) satisﬁes an analogous formula × · φ˜(0,z) = sup z w φ(0,w) = inf φ˜(ρ,z) = limφ˜(ρ,z) z Rd. (3.29) ρ ρ w Rd · − >0 0 ∀ ∈ ∈ ↓ Observe that, as in the (α-θ)-homogeneous case of Example 3.6 with α > 0,

+∞ if w = 0 φ˜(0,z) 0 φ(0,w) = 6 (3.30) ≡ ⇒ (0 if w = 0.

As in (2.4), we also introduce the recession functional

∞ 1 1 p 1 φ (ρ,w) = sup φ(λρ,λw) = lim φ(λρ,λw) = lim λ − φ(λρ,w). (3.31) λ λ λ +∞ λ λ +∞ >0 ↑ ↑ φ ∞ is still convex, p-homogeneous w.r.t. w, and l.s.c. with values in [0,+∞]; moreover, it is 1-homogeneous so that it can be expressed as

∞ ϕ (w) ρϕ∞ ρ ρ ∞ ρ p 1 = (w/ ) if = 0, φ (ρ,w) = − 6 (3.32) (+∞ if ρ = 0 and w = 0, 6 where ϕ∞ : Rd [0,+∞] is a convex and p-homogeneous function which is non degenerate, ∞ → ∞ i.e. ϕ (w) > 0 if w = 0. ϕ admits a dual representation, based on 6

∞ 1 1 ϕ˜ (z) := inf φ˜(λ,z) = lim φ˜(λρ,z). (3.33) λ>0 λ λ +∞ λ ↑ ϕ˜ ∞ is ﬁnite, convex, nonnegative, and q-homogeneous, so that ϕ˜ ∞(z)1/q is a seminorm, which does not vanish at z Rd if and only if ρ φ˜(ρ,z) has a linear growth when ρ +∞. It is easy to check that ∈ 7→ ↑

∞ ∞ ϕ (w)1/p = sup w z : ϕ˜ (z) 1 . (3.34) · ≤ n o In the case φ˜ has a sublinear growth w.r.t. ρ, as for (α-θ)-homogeneous functionals with α < 1 (see Example 3.6), we have in particular

∞ ∞ +∞ if w = 0, when ϕ˜ (z) 0, ϕ (w) = 6 (3.35) ≡ (0 if w = 0. 15

γ µ M+ Rd ν The action functional. Let , loc( ) be nonnegative Radon measures and let d d ∈ d ∈ Mloc(R ;R ) be a vector Radon measure on R . We assume that supp(µ),supp(ν) supp(γ), and we write their Lebesgue decomposition with respect to the reference measure⊂γ

µ := ργ + µ⊥, ν := wγ +ν ⊥. (3.36) We can always introduce a nonnegative Radon measure σ M+(Ω) such that µ = ρ σ ∈ ⊥ ⊥ ≪ σ,ν = w σ σ, e.g. σ := µ + ν . We can thus define the action functional ⊥ ⊥ ≪ ⊥ | ⊥| a ∞ ∞ Φ(µ,ν γ) = Φ (µ,ν γ) + Φ (µ,ν γ) := φ(ρ,w)dγ + φ (ρ⊥,w⊥)dσ. (3.37) Rd Rd | | | Z Z Observe that, being φ ∞ 1-homogeneous, this definition is independent of σ. We will also use a localized version of Φ: if B B(Rd) we set ∈ ∞ Φ(µ,ν γ,B) := φ(ρ,w)dγ + φ (ρ⊥,w⊥)dσ. (3.38) | ZB ZB Lemma 3.7 Let µ = ργ + µ⊥,ν = wγ +ν ⊥ be such that Φ(µ,ν γ) is finite. Then ν ⊥ = w µ µ and | ⊥ ⊥ ≪ ⊥ ∞ ∞ ∞ Φ (µ,ν γ) = ϕ (w⊥)dµ⊥, Φ(µ,ν γ) = φ(ρ,w)dγ + ϕ (w⊥)dµ⊥. (3.39) | Rd | Rd Rd Z Z Z Moreover, if φ˜ has a sublinear growth with respect to ρ (e.g. in the (α-θ)-homogeneous case of Example 3.6, with α < 1) then ϕ˜ ∞( ) 0 and · ≡ Φ(µ,ν) < +∞ ν = w γ γ, Φ(µ,ν) = Φa(µ,ν) = φ(ρ,w)dγ, (3.40) Rd ⇒ · ≪ Z independently on the singular part µ⊥. σ M+ Rd µ σ ν σ Φ∞ µ ν γ Proof Let ˜ loc( ) any measure such that ⊥ ˜ , ⊥ ˜ so that ( , ) can be represented∈ as ≪ | |≪ | ν ∞ ∞ dµ⊥ dν ⊥ Φ (µ,ν γ) = φ (ρ˜ ⊥,w˜ ⊥)dσ˜ , ρ˜ ⊥ = , w˜ ⊥ = . Rd dσ˜ dσ˜ | Z ∞ When Φ (µ,ν γ) < +∞, (3.32) yields w˜ ⊥(x) = 0 for σ˜ -a.e. x such that ρ˜ ⊥(x) = 0. It follows that | Φ(µ,ν) < +∞ ν ⊥ µ⊥, (3.41) ⇒ ≪ ν so that one can always choose σ˜ = µ⊥, ρ˜ ⊥ = 1, and decompose ν ⊥ as w⊥µ⊥ obtaining (3.39). (3.40) is then an immediate consequence of (3.35). ⊓⊔ Remark 3.8 When φ˜(0,z) 0 (e.g. in the (α-θ)-homogeneous case of Example 3.6, with α > 0) the density w of ν w.r.t.≡ γ vanishes if ρ vanishes, i.e. Φ(µ,ν γ) < +∞ w(x) = 0 if ρ(x) = 0, for γ-a.e. x Rd. (3.42) | ⇒ ∈ In particular ν a is absolutely continuous also with respect to µ. Applying Theorem 2.1 we immediately get Lemma 3.9 (Lower semicontinuity and approximation of the action functional) The action functional is lower semicontinuous with respect to weak∗ convergence of measures, i.e. if + d d d µn⇀∗µ, γn⇀∗γ weakly∗ in M (R ), ν n⇀∗ν in Mloc(R ;R ) as n +∞, loc ↑ then liminfΦ(µn,ν n γn) Φ(µ,ν γ). n ∞ | ≥ | ↑ 16

Equiintegrability estimate. We collect in this section some basic estimates on φ which will turn to be useful in the sequel. Let us ﬁrst introduce the notation

1/q 1/p 1 z := φ˜(1,z) , w := φ(1,w) , η− z z η z , (3.43) k k∗ k k | | ≤ k k∗ ≤ | | Γφ := (a,b) : sup φ˜(ρ,z) a + bρ , h(ρ) := inf a + bρ : (a,b) Γφ , (3.44) z =1 ≤ ∈ n k k∗ o n o H(s,ρ) := sh(ρ/s) = inf as + bρ : (a,b) Γφ . (3.45) ∈ n o Observe that h is a concave increasing function deﬁned in [0,+∞), satisfying, in the homogeneous case h(ρ) = h(ρ) = ρα . It provides the bounds

φ˜(ρ,z) h(ρ) z q, w h(ρ)1/qφ(ρ,w)1/p, ≤ k k∗ k k ≤ ∞ ∞ q ∞ 1/q ∞ 1/p ∞ 1 (3.46) ϕ˜ (z) h z , w h ϕ (w) , if h := lim λ − h(λ) > 0. ≤ k k∗ k k ≤ λ +∞ ↑ Observe that when h∞ = 0 then ϕ˜ ∞ 0 and ϕ∞(w) is given by (3.35). ≡ Proposition 3.10 (Integrability estimates) Let ζ be a nonnegative Borel function such that

µ(ζ q) := ζ q dµ and γ(ζ q) := ζ q dγ are ﬁnite, Rd Rd Z Z

and let Z := x Rd : ζ(x) > 0 . If Φ(µ,ν γ) < +∞ we have ∈ | ζ(x)d ν (x) Φ1/p µ,ν γ,Z H1/q γ(ζ q), µ(ζ q) . (3.47) Rd Z k k ≤ | In particular, for every Borel set A B(Rd) we have ∈

ν (A) Φ1/p µ,ν γ,A H1/q(γ(A), µ(A) . (3.48) k k ≤ | ∞ Proof It is sufﬁcient to prove (3.47). Observe that if (a,b) Γφ then a 0, and h b so that by (3.46) we have ∈ ≥ ≤

ζ(x)d ν (x) ζ w dγ + ζ w⊥ dµ⊥ Rd k k ≤ Z k k Z k k Z Z Z 1/p 1/q 1/p 1/q q ∞ ∞ q φ(ρ,w)dγ ζ h(ρ)dγ + ϕ (w⊥)dµ⊥ h ζ dµ⊥ ≤ Z Z Z Z Z Z Z Z 1/p 1/q Φ(µ,ν γ,Z) a ζ q dγ + b ζ q dµ , ≤ | Rd Rd Z Z

Taking the inﬁmum of the last term over all the couples (a,b) Γφ we obtain (3.48). ∈ ⊓⊔ 17

4 Measure valued solutions of the continuity equation in Rd

In this section we collect some results on the continuity equation

d ∂t µt + ∇ ν t = 0 in R (0,T), (4.1) · × which we will need in the sequel. Here µt ,ν t are Borel families of measures (see e.g. [3]) in M+ Rd M Rd Rd loc( ) and loc( ; ) respectively, deﬁned for t in the open interval (0,T), such that T T µt (BR)dt < +∞, VR := ν t (BR)dt < +∞ R > 0, (4.2) Z0 Z0 | | ∀ and we suppose that (4.1) holds in the sense of distributions, i.e.

T T ∂t ζ(x,t)dµt (x)dt + ∇xζ(x,t) dν t(x)dt = 0 (4.3) 0 Rd 0 Rd · Z Z Z Z ζ 1 Rd for every Cc ( (0,T )). Thanks to the disintegration theorem [14, 4, III-70], we can ν ∈ × ν T ν M Rd Rd identify ( t )t (0,T) with the measure = 0 t dt loc( (0,T); ) deﬁned by the formula ∈ ∈ × R T ν ζ ζ ν ζ 0 Rd Rd , = (x,t) d t (x) dt Cc ( (0,T); ). (4.4) h i 0 Rd · ∀ ∈ × Z Z

4.1 Preliminaries

Let us ﬁrst adapt the results of [3, Chap. 8] (concerning a family of probability measures µt ) to the more general case of Radon measures. First of all we recall some (technical) preliminaries.

Lemma 4.1 (Continuous representative) Let µt ,ν t be Borel families of measures satisfying (4.2) and (4.3). Then there exists a unique weakly∗ continuous curve t [0,T] µ˜t M+ Rd µ µ L 1 ζ 1 Rd ∈ 7→ ∈ loc( ) such that t = ˜t for -a.e. t (0,T); if Cc ( [0,T]) and t1 t2 [0,T], we have ∈ ∈ × ≤ ∈ t t ζ µ ζ µ 2 ∂ ζ µ 2 ∇ζ ν t2 d ˜t2 t1 d ˜t1 = t d t (x)dt + d t (x)dt, (4.5) Rd − Rd t Rd t Rd · Z Z Z 1 Z Z 1 Z and the mass of µ˜t can be uniformly bounded by 1 sup µ˜t (BR) µ˜ s(B2R) + 2R− V2R s [0,T]. (4.6) t [0,T ] ≤ ∀ ∈ ∈ d 1 Moreover, if µ˜s(R ) < +∞ for some s [0,T] and limR +∞ R− VR = 0, then the total mass d ∈ ↑ µ˜t (R ) is (ﬁnite and) constant. ζ η ζ η ∞ ζ ∞ Rd ζ Proof Let us take (x,t) = (t) (x), Cc (0,T) and Cc ( ) with supp BR; we have ∈ ∈ ⊂ T T η′(t) ζ(x)dµt (x) dt = η(t) ∇ζ(x) dν t (x) dt, − 0 Rd 0 Rd · Z Z Z Z 1,1 so that the map t µt (ζ) = Rd ζ dµt belongs to W (0,T) with distributional derivative 7→ R 1 µ˙t (ζ) = ∇ζ(x) dν t (x) for L -a.e. t (0,T), (4.7) Rd · ∈ Z 18

satisfying

T µ˙t (ζ) VR(t)sup ∇ζ , VR(t) := ν t (BR), VR(t)dt = VR < +∞. (4.8) | | ≤ d | | | | 0 R Z

1 If Lζ is the set of its Lebesgue points, we know that L ((0,T) Lζ ) = 0. Let us now take n ∞ \∞ an increasing sequence Rn := 2 + and countable sets Zn Cc (BRn ) which are dense in 1 ζ 1 Rd ↑ζ ⊂1 C0 (BRn ) := C ( ) : supp( ) BRn , the closure of Cc (BRn ) with respect the usual 1 ζ{ ∈ ζ ∇ζ ⊂ } C norm C1 = supRd ( , ). We also set LZ := n N,ζ Zn Lζ . The restriction of the µ k k | | | | ∩ ∈ ∈ 1 curve to LZ provides a uniformly continuous family of functionals on each space C0 (BRn ), since (4.8) shows

t µ ζ µ ζ ζ λ λ ζ t ( ) s( ) C1 VRn ( )d s,t LZ Zn. | − | ≤ k k Zs ∀ ∈ ∀ ∈ Therefore, for every n N it can be extended in a unique way to a continuous curve µn 1 ∈ ˜t t [0,T ] in [C0(BRn )]′ which is uniformly bounded and satisﬁes the compatibility condition{ } ∈ m n 1 µ˜ (ζ) = µ˜ (ζ) if m n and ζ C (BR ). (4.9) t ≤ ∈ c m If ζ C1(Rd) we can thus deﬁne ∈ c n µ˜t (ζ) := µ˜ (ζ) for every n N such that supp(ζ) BR . (4.10) t ∈ ⊂ n µ N If we show that t (BRn ) t LZ is uniformly bounded for every n , the extension provides { M+} ∈Rd ∈ a continuous curve in loc( ). To this aim, let us consider nonnegative, smooth functions

d k ζk : R [0,1], such that ζk(x) := ζ0(x/2 ), (4.11a) → k k+1 k ζk(x) = 1 if x 2 , ζk(x) = 0 if x 2 , ∇ζk(x) A2− , (4.11b) | | ≤ | | ≥ | | ≤

for some constant A > 1. It is not restrictive to suppose that ζk Zk 1. Applying the previous ∈ + formula (4.7), for t, s LZ we have ∈ T µ ζ µ ζ 1 k ν k t ( k) s( k) ak := 2 − r B2Rk BRk dr A2− V2Rk . (4.12) | − | ≤ Z0 | | \ ≤ It follows that

k k µt (BR ) µt (ζk) µs(ζk) + A2− V2R µs(B2R ) + A2− V2R t LZ . (4.13) k ≤ ≤ k ≤ k k ∀ ∈ Integrating with respect to s we end up with the uniform bound

T µ k µ ∞ t (BRk ) A2− VRk+1 + s(B2Rk )ds < + t LZ. ≤ 0 ∀ ∈ Z

Observe that the extension µ˜t satisﬁes (4.13) (and therefore, in a completely analogous way, (4.6)) and (4.12) for every s,t [0,T]. ∈ 1 d ∞ Now we show (4.5). Let us choose ζ C (R [0,T]) and ηε C (t1,t2) such that ∈ c × ∈ c η η χ η δ δ 0 ε (t) 1, lim ε (t) = (t ,t )(t) t [0,T], lim ε′ = t t ≤ ≤ ε 0 1 2 ∀ ∈ ε 0 1 − 2 ↓ ↓ 19

in the duality with continuous functions in [0,T]. We get

T T 0 = ∂t (ηε ζ)dµt (x)dt + ∇x(ηε ζ) dν t dt Rd Rd Z0 Z Z0 Z · T T T = ηε (t) ∂t ζ dµt dt + ηε (t) ∇xζ dν t dt + ηε′ (t) ζ dµ˜t dt. Rd Rd Rd Z0 Z Z0 Z · Z0 Z Passing to the limit as ε vanishes and invoking the continuity of µ˜t , we get (4.5). 1 Finally, if limR +∞ R− VR = 0 we can pass to the limit as Rk +∞ in the inequality (4.12), which also holds↑ for every t,s [0,T] if we replace µ by µ˜ , by↑ choosing s so that ∈ d m := µ˜s(R ) = lim µ˜s(ζk) < +∞. k +∞ ↑ d It follows that µ˜t (R ) = limk +∞ µ˜ s(ζk) = m for every t [0,T]. ↑ ∈ ⊓⊔ Thanks to Lemma 4.1 we can introduce the following class of solutions of the continuity equation.

Deﬁnition 4.2 (Solutions of the continuity equation) We denote by CE(0,T) the set of µ ν time dependent measures ( t )t [0,T],( t )t (0,T) such that ∈ ∈ + d 1. t µt is weakly continuous in M (R ) (in particular, sup µt (BR) < +∞ for 7→ ∗ loc t [0,T ] every R > 0), ∈ T ν ν ∞ 2. ( t )t (0,T) is a Borel family with t (BR)dt < + R > 0; ∈ 0 | | ∀ 3. (µ,ν) is a distributional solutionZ of (4.1).

CE(0,T;σ η) denotes the subset of (µ,ν) CE(0,T) such that µ0 = σ, µ1 = η. → ∈ Solutions of the continuity equation can be rescaled in time:

Lemma 4.3 (Time rescaling) Let t : s [0,T ′] t(s) [0,T] be a strictly increasing ab- ∈ → ∈ 1 solutely continuous map with absolutely continuous inverse s := t− . Then (µ,ν) is a distributional solution of (4.1) if and only if

µˆ := µ t, νˆ := t′ ν t , is a distributional solution of (4.1) on (0,T ′). ◦ ◦ We refer to [3, Lemma 8.1.3] for the proof. The proof of the next lemma follows directly from (4.5). µi ν i CE µ1 µ2 Lemma 4.4 (Glueing solutions) Let ( , ) (0,Ti),i = 1,2, with T = 0 . Then the µ ν ∈ 1 new family ( t , t )t (0,T +T ) deﬁned as ∈ 1 2 1 1 µ if 0 t T1 ν if 0 t T1 µ := t ≤ ≤ ν := t ≤ ≤ (4.14) t µ2 t ν 2 ( t T if T1 t T1 + T2 ( t T if T1 t T1 + T2 − 1 ≤ ≤ − 1 ≤ ≤ belongs to CE(0,T1 + T2).

Lemma 4.5 (Compactness for solutions of the continuity equation (I)) Let (µn,ν n) be a sequence in CE(0,T) such that µn ∞ 1. forsome s [0,T] supn N s (BR) < + R > 0; ∈ ∈ n ∀ 2. the sequence of maps t ν (BR) is equiintegrable in (0,T), for every R > 0. 7→ | t | 20

Then there exists a subsequence (still indexed by n) and a couple (µt ,ν t ) CE(0,T) such that (recall (4.4)) ∈ µn µ M+ Rd t ⇀∗ t weakly∗ in ( ) t [0,T], loc ∀ ∈ (4.15) n d d ν ⇀∗ν weakly∗ in Mloc(R (0,T);R ). × (4.15) yields in particular

T T Φ µ ν γ Φ µn ν n γn ( t , t )dt liminf ( t , t )dt (4.16) 0 | ≤ n +∞ 0 | Z ↑ Z

γn γ M+ Rd Φ for every sequence of Radon measures ⇀∗ in loc( ), where is an integral functional as in (3.37).

ν n T ν n µn Proof Since := 0 t dt and s have total variation uniformly bounded on each com- Rd µn ν n pact subset of [0,T], we can extract a subsequence (still denoted by s , ) such that n ×dR n d d µ ⇀ µs in Mloc(R ) and ν ⇀ ν in Mloc(R [0,T];R ). The estimate (4.6) shows that s ∗ ∗ ×

µn ∞ sup t (BR) < + t [0,T], R > 0. (4.17) n N ∀ ∈ ∈

The equiintegrability condition on ν n shows that ν satisﬁes

1 ν (BR I) = mR(t)dt I B(0,T), R > 0, for some mR L (0,T), | | × ZI ∀ ∈ ∈

ν T ν so that by the disintegration theorem we can represent it as = 0 t for a Borel family ν ζ 1 Rd t t (0,T ) still satisfying (4.2). Let us now consider a function Cc ( ) and for a given { } ∈ χ∈R interval I =[t0,t1] [0,T] the time dependent function ζ (t,x) := I (t)∇ζ(x). Since the dis- ⊂ d continuity set of ζ is concentrated on N = R t0,t1 and ν (N) = 0, general convergence theorems (see e.g. [3, Prop. 5.1.10] yields ×{ } | |

lim ∇ζ x dν n x dt lim ζ dν n t x ∞ ( ) t ( ) = ∞ ( , ) n I Rd · n Rd (0,T ) · → Z Z → Z × (4.18) = ζ dν (t,x) = ∇ζ(x) dν t(x)dt. Rd (0,T ) · I Rd · Z × Z Z

Applying (4.5) with ζ(t,x) := ζ(x) and t0 := s and the estimate (4.17) we thus obtain the n + d weak convergence of µ to a measure µt M (R ) for every t [0,T]. It is immediate to t ∈ ∈ check that the couple (µt,ν t ) belongs to CE(0,T). (4.16) follows now by the representation

T T Φ µ ν γ Φ µ ν γ µ µ γ γ L 1 M+ Rd ( t , t )dt = ( , ¯), := t dt, ¯ = loc( (0,T)) Z0 | | Z0 ⊗ ∈ × and the lower semicontinuity property stated in Theorem 2.1. ⊓⊔ 21

4.2 Solutions of the continuity equation with ﬁnite Φ-energy

For all this section we will assume that φ : (0,+∞) Rd (0,+∞) is an admissible action ×∞ γ→ M+ Rd density function as in (3.2a,b,c) for some p (1,+ ), loc( ) is a given reference Radon measure, and Φ is the corresponding∈ integral functional∈ as in (3.37). We want to study the properties of measure valued solutions (µ,ν) of the continuity equation (4.1) with finite Φ-energy T E := Φ(µt ,ν t γ)dt < +∞. (4.19) 0 | Z We denote by CEφ,γ (0,T) the subset of CE(0,T) whose elements (µ,ν) satisfies (4.19). µ M+ Rd µ ν Remark 4.6 If ( t )t [0,T ] is weakly∗ continuous in loc( ) and (4.19) holds, then t , t ∈ µ µ also satisfy (4.2): in fact, the weak∗ continuity of t yields for every R > 0 supt [0,T ] t (BR) = ∈ MR < +∞, and the estimate (3.48) yields (recall (3.43)) T 1/q 1/p 1/q VR η ν t (BR)dt η T E H(γ(BR),MR) < +∞. (4.20) ≤ 0 k k ≤ Z Recalling that the function h is defined by (3.44), we also introduce the concave function s 1 1 ω(s) := dr, ω(0) = 0, ω′(s) = , lim ω(s) = +∞. (4.21) 0 h(r)1/q h(s)1/q s ∞ Z → In the homogeneous case φ(ρ,z) = ρα z q we have k k∗ s α/q q 1 α/q p θ/p ω(s) = r− dr = s − = s . (4.22) 0 q α θ Z − For given nonnegative ζ C1(Rd) and µ M+ (Rd) we will use the short notation ∈ c ∈ loc d p Z := supp(Dζ) R , Gp(ζ) := ζ dγ, D(ζ) := sup Dζ . (4.23) ∗ ⊂ ZZ Rd k k Theorem 4.7 Let ζ C1(Rd) be a nonnegative function with Z,G(ζ),D(ζ) defined as in ∈ c (4.23), and let µ,ν CEφ γ (0,T). Setting ∈ , T EZ := Φ(µt ,ν t γ,Z)dt E < +∞, (4.24) 0 | ≤ Z we have d p 1/p p 1/q µt (ζ ) pD(ζ)Φ(µt,ν t γ,Z) H Gp(ζ), µt(ζ ) . (4.25) dt ≤ | C h In particular, there exists a constant 1 > 0 only depending (in a monotone way) on , p,T such that µ ζ p C µ ζ p ζ ζ 1/q 1/p p ζ sup t ( ) 1 0( ) + D( )Gp( ) EZ + D ( )EZ . (4.26) t [0,T ] ≤ ∈ Moreover, if Gp(ζ) > 0, pD(ζ) d ω (µ (ζ p)/G (ζ)) Φ(µ ,ν γ,Z)1/p for a.e. t (0,T). (4.27) dt t p 1/p t t ≤ Gp(ζ) | ∈

In particular, in the (α-θ)-homogeneous case, for every 0 s t T we have ≤ ≤ ≤ t ζ θ ζ θ θ ζ ζ θ 1 Φ µ ν γ 1/p Lp(µ ) Lp(µ ) D( ) Lp−(γ,Z) ( r, r ,Z) dr. (4.28) k k t − k k s ≤ k k s | Z

p Proof Setting mt := µt (ζ ), G = Gp(ζ), D = D(ζ) we easily have by (3.47)

d d p p 1 1/p 1/q mt = ζ dµt = p ζ − ∇ζ dν t pDΦ(µt ,ν t γ,Z) H G,mt , dt dt Rd Z ZZ · ≤ | p 1 q p since (ζ − ) = ζ . Since H G,mt ) = Gh(mt/G), we get

1/q d 1/q 1/p h− (mt/G) mt pDG Φ(µt ,ν t γ,Z) . dt ≤ | d ω h 1/q Recalling that dr (r) = − (r) we get (4.27). Γ In order to prove (4.26) we set M := supt [0,T ] mt and we choose constants (a,b) φ ; integrating (4.25) we get ∈ ∈

1/q 1/q 1/p 1/q 1/p sup mt m0 pDT aG EZ + bM EZ . (4.29) t [0,T ] − ≤ ∈

By using the inequality xy p 1xp + q 1yq we obtain ≤ − − 1/q 1/p 1 p 1 p p/q M m0 + pD aTG E + M + p − D bT EZ (4.30) ≤ Z q 1/q p 1 p/q which yields (4.26) with C1 := pmax 1, p(aT ) , p − (bT) . Finally, let us assume that φ satisﬁes the (α-θ)-homogeneity condition, so that ω(s) = p θ/p θ s as in (4.22). It follows that

1 p θ θ ω(G− mt ) = ζ p µ ζ −p . (4.31) θ k kL ( t )k kL (γ,Z) Integrating (4.27) we conclude. ⊓⊔ We extend the deﬁnition of mr(µ) also for negative values of r by setting

r r m˜ r(µ) := µ(B1) + x dµ(x) = 1 x dµ(x) r R. (4.32) Rd B | | Rd ∨| | ∀ ∈ Z \ 1 Z d Notice thatm ˜ 0(µ) = µ(R ) and mr(µ) m˜ r(µ) µ(B1) + mr(µ) when r > 0. ≤ ≤ Theorem 4.8 Let us assume that m˜ r(γ) < +∞ for some r p and let (µ,ν) CEφ γ (0,T) ≤ ∈ , satisfy (4.19). For every δ 1 + r/q, if m˜ δ (µ0) < +∞ then also m˜ δ (µt ) < +∞ and there ≤ exists a constant C2 only depending in a monotone way on h, p,T,A, δ such that | | 1/q 1/p m˜ δ (µt ) C2 m˜ δ (µ0) + m˜ r(γ) E + E . (4.33) ≤ d d Moreover, if r q and µ0(R ) < +∞, then µt (R ) is ﬁnite and constant for every t [0,T]. ≥ − ∈ Proof Let us ﬁrst set nr Kn := 2 γ(B n+1 B2n ) (4.34) 2 \ observing that +∞ r Kn ∑ Kj 2 − m˜ r(γ), limsupKn = 0. (4.35) ≤ j=0 ≤ n +∞ ↑ ∞ d We consider the usual cutoff functions ζn C (R ) as in (4.11a,b) and we set ∈ c ζ ζ n ζ γ nr Dn = D( n) = sup D n A2− , Gn = Gp( n) (B2n+1 B2n ) = 2− Kn. (4.36) k k∗ ≤ ≤ \ 23

By (4.26) we obtain

µ C µ n(1+r/q) 1/q 1/q p np sup t (B2n ) 1 0(B2n+1 ) + A2− Kn E + A 2− E ; (4.37) t [0,T ] ≤ ∈ d d in particular, if r q and µ0(R ) < +∞, we can derive the uniform upper bound µt (R ) d ≥ − d ≤ C1 µ0(R ) letting n +∞. We can then deduce that µt (R ) is constant by applying the ↑ estimate (4.25), which yields after an integration in time and for every (a,b) Γφ ∈ µ ζ p µ ζ p 1/q 1/p n nr C µ Rd 1/q sup t ( n ) 0( n ) pAT E 2− a2− Kn + b 1 0( ) . t [0,T ] − ≤ ∈

In order to show (4.33), we argue as before, by introducing the new family of test functions n ∞ d induced by υn(x) := υ0(x/2 ) C (R ) ∈ c υ (x) 1 if2n x 2n+1, 0 υ 1 n Dυ A2 n (4.38) n , ≡ ≤| n| ≤1 n+2 n − . ≤ ≤ υn(x) 0 if x 2 or x 2 , k k∗ ≤ ( ≡ | | ≤ − | | ≥ +∞ p Observe that 1 ∑ υn(x) 3 and for some constant Aδ > 1 ≤ n=1 ≤ +∞ 1 δ δn p δ d A−δ x ∑ 2 υn(x) Aδ x x R , x 2. (4.39) | | ≤ n=1 ≤ | | ∀ ∈ | | ≥ υ n As before, setting Kn′ := Kn+1 + Kn 1, we have D( n) A2− and − ≤ υ (n+1)r (n 1)r r nr Gp( n) 2− Kn+1 + 2− − Kn 1 2| | 2− Kn′ . (4.40) ≤ − ≤ Applying (4.26) we get for every t [0,T] ∈ nδ p nδ p (δ 1 r/q)n 1/q 1/p p (δ p)n 2 µt (υ ) C1 2 µ0(υ ) + A2 − − (K′ ) (E′ ) + A 2 − E′ , n ≤ n n n n where T Φ µ ν γ En := ( t , t ,B2n+1 B2n )dt, En′ := En+1 + En 1. (4.41) − Z0 | \ Since δ 1 + r/q and δ p, summing up with respect to n and recalling (4.37) we get ≤ ≤ 1/q 1/p m˜ δ (µt ) C2 m˜ δ (µ0)+(m˜ r(γ)) E + E . (4.42) ≤ ⊓⊔ In the the θ-homogeneous case we have a more reﬁned estimate:

Theorem 4.9 Let us assume that φ is θ-homogeneous for some θ (1, p], the measure γ ∈ satisﬁes the r-moment condition m˜ r(γ) < +∞, and let (µ,ν) CEφ,γ (0,T) satisfy (4.19). ¯ 1 1 ∈ For every δ δ := p +(1 )r, if m˜ δ (µ0) < +∞ then m˜ δ (µt ) is ﬁnite and there exists a ≤ θ − θ constant C3 > 0 such that

1 1/θ 1/θ m˜ δ (µt ) C3 m˜ δ (µ0) + m˜ r(γ) − E . (4.43) ≤ ¯ d d Moreover, if δ 0 (i.e. r p/(θ 1)) and µ0(R ) < +∞ then µt (R ) is ﬁnite and constant ≥ ≥ − − for t [0,T]. ∈ 24

Proof We argue as in the proof of Theorem (4.9), keeping the same notation and using the crucial estimate (4.28). If ζn are the test functions of (4.11a,b),

ζ θ 1 (θ 1)/p (4.36) nr(θ 1)/p (θ 1)/p n p−γ = Gn − = 2− − Kn − , (4.44) k kL ( ,Zn) so that, since δθ¯ /p = 1 +(θ 1)r/p, (4.28) yields − p θ/p p θ/p δ¯ θn/p (θ 1)/p 1/p µt (ζ ) µ0(ζ ) Aθ2− Kn − E . (4.45) n − n ≤

¯ d d Since δ 0, passing to the limit as n ∞ and recalling (4.35), we get µt (R ) µ0(R ). ≥ ↑ ≡ Concerning the moment estimate, we replace ζn by υn, deﬁned by in (4.38), obtaining

p θ/p p θ/p δ¯ θn/p (θ 1)/p 1/p µt (υ ) µ0(υ ) C3 12− K′ − E′ , (4.46) n − n ≤ . n n

and therefore p p δ¯n 1 1/θ 1/θ µt (υ ) C3 2 µ0(υ ) + 2− K′ − E′ . (4.47) n ≤ . n n n Multiplying this inequality by 2nδ , summing up w.r.t. n, and recalling (4.39), we obtain

1 1/θ 1/θ m˜ δ (µt ) C3 m˜ δ (µ0) + m˜ r(γ) − E . (4.48) ≤ ⊓⊔ Corollary 4.10 (Compactness for solutions of the continuity equation (II)) Let (µn,ν n) CE γn γ M+ Rd be a sequence in φ,γ (0,T) and let ⇀∗ in loc( ) such that

T µn ∞ Φ µn ν n γn ∞ sup 0 (BR) < + R > 0, sup ( t , t )dt < + . (4.49) n N ∀ n N 0 | ∈ ∈ Z Then conditions 1. and 2. of Lemma 4.5 are satisﬁed and therefore there exists a subsequence (still indexed by n) and a couple (µt,ν t ) CEφ γ (0,T) such that ∈ , µn µ M+ Rd t ⇀∗ t weakly∗ in ( ) t [0,T], loc ∀ ∈ (4.50) n d d ν ⇀∗ν weakly∗ in Mloc(R (0,T);R ), × T T Φ µ ν γ Φ µn ν n γn ( t , t )dt liminf ( t , t )dt. (4.51) 0 | ≤ n +∞ 0 | Z ↑ Z µn Rd µ Rd m γn ∞ κ κ Suppose moreover that 0 ( ) 0( ) and supn ˜ κ ( ) < + where = q or = θ θ → −µn Rd p/( 1) in the -homogeneous case, then (along the same subsequence) t ( ) − d − → µt (R ) for every t [0,T]. ∈ γn ∞ δ Proof Since PR := supn (BR) < + for every R > 0, the estimate (4.33) for = 0 and µn ∞ the assumption (4.49) show that MR = supn N,t [0,T ] t (BR) < + for every R > 0. We can n ∈ ∈ therefore obtain a bound of ν (BR) by (3.48), which yields k t k n 1/q 1/p ν (BR) H(PR,MR) Φ(µt ,ν t γ) , k t k ≤ | ν n p so that the maps t t (BR) are uniformly bounded by a function in L (0,T). The last assertion follows by7→ the k factk that t µn(Rd) is independent of time, thanks to Theorem 4.9 7→ t (in the (α-θ)-homogeneous case) or Theorem 4.8 (for general density functions φ). ⊓⊔ 25

5 The (φ-γ)-weighted Wasserstein distance

As we already mentioned in the Introduction, BENAMOU-BRENIER [7] showed that the Wasserstein distance Wp (1.1) can be equivalently characterized by a “dynamic” point of view through (1.15), involving the 1-homogeneous action functional (1.13). The same approach can be applied to arbitrary action functionals.

γ M+ Rd Definition 5.1 (Weighted Wasserstein distances) Let loc( ) be a fixed reference measure and φ : (0,+∞) Rd [0,+∞) a function satisfying∈ Conditions (3.2a,b,c). The × → + d (φ,γ)-Wasserstein (pseudo-) distance between µ0, µ1 M (R ) is defined as ∈ loc 1 Wp µ µ Φ µ ν γ µ ν CE µ µ φ,γ ( 0, 1) := inf ( t , t )dt : ( , ) (0,1; 0 1) . (5.1) 0 | ∈ → nZ o + d We denote by Mφ γ [µ0] the set of all the measures µ M (R ) which are at finite Wφ γ - , ∈ loc , distance from µ0.

α α p Remark 5.2 Let us recall the notation Wp α;γ of (1.23) in the case φp α (ρ,w) = ρ w/ρ . , , | | When α = 0 we ﬁnd the dual homogeneous Sobolev (pseudo-)distance (1.7) and in the case α = 1 and supp(γ) = Rd we get the usual Wasserstein distance:

µ0 µ1 1,p = Wp,0;γ (µ0, µ1), Wp(µ0, µ1) = Wp,1;γ (µ0, µ1). k − kW˙ γ− Remark 5.3 Taking into account Lemma 4.3, a linear time rescaling shows that

T p µ µ p 1 Φ µ ν γ µ ν CE µ µ Wφ,γ ( 0, T ) := inf T − ( t , t )dt : ( , ) (0,T; 0 T ) . (5.2) 0 | ∈ → n Z o Theorem 5.4 (Existence of minimizers) Whenever the infimum in (5.1) is a finite value W < +∞, it is attained by a curve (µ,ν) CEφ γ (0,1) such that ∈ , 1 Φ(µt ,ν t γ) = W for L -a.e. t (0,1). (5.3) | ∈ µ The curve ( t)t [0,1] associated to a minimum for (5.1) is a constant speed mimimal geodesic ∈ for Wφ,γ since it satisfies

Wφ γ (µs, µt ) = t s Wφ γ (µ0, µ1) s,t [0,1]. (5.4) , | − | , ∀ ∈ We have also the equivalent characterization

T 1/p Wφ,γ (σ,η) = inf Φ(µt ,ν t γ) dt : (µ,ν) CE(0,T;σ η) . (5.5) 0 | ∈ → nZ o Proof When Wφ,γ (µ0, µ1) < +∞, Corollary 4.10 immediately yields the existence of a minimizing curve (µ,ν). Just for the proof of (5.5), let us denote by W¯ φ,γ (σ,η) the infimum of the right-hand side of (5.5). Hölder inequality immediately shows that Wφ γ (σ,η) , ≥ W¯ φ,γ (σ,η). In order to prove the opposite inequality, we argue as in [3, Lemma 1.1.4], defining for (µ,ν) CE(0,T;σ η) ∈ → t 1/p sε (t) := ε + Φ(µr,ν r γ) dr, t [0,T]; (5.6) 0 | ∈ Z 26

sε is strictly increasing with sε ε, sε (0,T)=(0,Sε ) with Sε := sε (T), so that its inverse ′ ≥ map tε : [0,Sε ] [0,T] is well deﬁned and Lipschitz continuous, with → 1/p − tε′ sε = ε + Φ(µt ,ν t ) a.e. in (0,T). (5.7) ◦ ε ε ε ε If µˆ = µ tε ,νˆ := tε ν tε , we know that (µˆ ,νˆ ) CE(0,Sε ;σ η) so that ◦ ′ ◦ ∈ → Sε T Φ µε ν ε γ 1/p p p 1 ε ε p 1 ( t , t ) W σ η Sε − Φ µˆ νˆ γ ds Sε − ε Φ µ ν dt φ,γ ( , ) ( s , s ) = ε Φ µε ν|ε γ + ( t , t ) , ≤ 0 | 0 + ( t , t ) Z Z | p the latter integral being less than Sε . Passing to the limit as ε 0, we get ↓ T 1/p Wφ,γ (σ,η) Φ(µt ,ν t γ) dt (µ,ν) CE(0,T;σ η), (5.8) ≤ Z0 | ∀ ∈ →

and therefore Wφ,γ (σ,η) W¯ φ,γ (σ,η). If (µ,ν) CEφ,γ (0,1; µ0 µ1) is a minimizer of (5.1), then (5.8) yields ≤ ∈ →

1 1/p 1 1/p 1/p W = Wφ,γ (µ0, µ1) = Φ(µt ,ν t γ)dt = Φ(µt ,ν t γ) dt, 0 | 0 | Z Z so that (5.3) holds. ⊓⊔

5.1 Topological properties

Theorem 5.5 (Distance and weak convergence) The functional Wφ,γ is a (pseudo)-distance M+ Rd on loc( ) which induces a stronger topology than the weak∗ one. Bounded sets with respect to Wφ,γ are weakly∗ relatively compact.

Proof It is immediate to check that Wφ γ is symmetric (since φ(ρ, w) = φ(ρ,w)) and , − Wφ,γ (σ,η) = 0 σ η. The triangular inequality follows as well from the characterization (5.5) and the⇒ gluing≡ Lemma 4.4. From (4.27) (keeping the same notation (4.23)) and (5.5) we immediately get for every + d 1 d µ0, µ1 M (R ) and nonnegative ζ C (R ) with ζ p γ > 0 ∈ loc ∈ c k kL ( ) pD ζ ω µ ζ p ζ ω µ ζ p ζ ( ) W σ η 1( )/Gp( ) 0( )/Gp( ) ζ φ,γ ( , ), − ≤ Lp(γ) k k which shows the last assertion, since ω is strictly increasing and the set

p 1 d ζ : ζ C (R ), ζ 0, ζ p γ > 0 ∈ c ≥ k kL ( ) is dense in the space of nonnegative continuous functions with compact support (endowed with the uniform topology). ⊓⊔

Theorem 5.6 (Lower semicontinuity) The map (µ0, µ1) Wφ,γ (µ0, µ1) is lower semi- M+ Rd 7→ continuous with respect to weak∗ convergence in loc( ). More generally, suppose that γn γ M+ Rd φ n φ ⇀∗ in loc( ), is monotonically increasing w.r.t. n and pointwise converging to , n n + d and µ ⇀ µ0, µ ⇀ µ1 in M (R ) as n +∞. Then 0 ∗ 1 ∗ loc ↑ W µn µn W µ µ liminf φ n,γn ( 0 , 1 ) φ,γ ( 0, 1). (5.9) n +∞ ≥ ↑ 27

W µn µn ∞ Proof It is not restrictive to assume that φ n,γn ( 0 , 1 ) < S < + , so that we can ﬁnd a n n n n sequence (µ ,ν ) CEφ n γn (0,1; µ µ ) such that ∈ , 0 → 1 Φm(µn,ν n γn) S a.e. in (0,1), m n N, (5.10) t t | ≤ ∀ ≤ ∈ where Φm denotes the integral functional associated to φ m. We can apply Theorem 4.10 and we can extract a suitable subsequence (still denoted bu µn,ν n) and a limit curve (µ,ν) ∈ CEφ γ (0,1; µ0 µ1) such that (4.50) holds. We eventually have , → 1 Wp µ µ Φm µ ν γ φ m,γ ( 0, 1) ( t , t )dt S. (5.11) ≤ Z0 | ≤ Passing to the limit w.r.t. m +∞ we conclude. ↑ ⊓⊔ + Theorem 5.7 (Completeness) For every σ M (Rd) the space Mφ γ [σ] endowed with ∈ loc , the distance Wφ,γ is complete.

Proof Let (µn)n N be a Cauchy sequence in Mφ,γ [σ] w.r.t. the distance Wφ,γ ; in particular, µ ∈ µ ( n) is bounded so that we can extract a suitable convergence subsequence nk weakly∗ µ M+ Rd W µ µ converging to ∞ in loc( ). Thanks to Theorem 5.6 we easily get φ,γ ( m, ∞) W µ µ ≤ liminfk ∞ φ,γ ( m, nk ), and therefore, taking into account the Cauchy condition, → W µ µ W µ µ µ µ limsupm ∞ φ,γ ( m, ∞) limsupn,m ∞ φ,γ ( m, n) = 0 so that n converges to ∞. → ≤ → ⊓⊔ Let us now consider the case of measures with ﬁnite mass (just to ﬁx the constant, probability measures in P(Rd )). We introduce the parameter p q = if φ is (α-θ)-homogeneous, κ := θ 1 1 α (5.12)  p− − p 1 = q otherwise.  − Theorem 5.8 (Distance and total mass) Let us assume that m˜ κ (γ) < +∞ and let us sup-  − pose that σ P(Rd). Then Mφ γ [σ] P(Rd), the weighted Wasserstein distance Wφ γ is ∈ , ⊂ , stronger than the narrow convergence in P(Rd), and P(Rd ) endowed with the (pseudo-) distance Wφ,γ is a complete (pseudo-)metric space.

Proof If η Mφ,γ [σ] then Theorem 4.9 (in the θ-homogeneous case) or 4.8 (in the gen- ∈ d d d eral case) yields η(R ) = σ(R ) = 1, so that Mφ,γ [σ] P(R ). Since the narrow topology d ⊂ coincide with the weak∗ one in P(R ), Theorem 5.5 proves the second statement. The completeness of P(Rd) with respect to the (pseudo) distance Wφ γ follows by Theorem 5.7. , ⊓⊔ We can also prove some useful moment estimates.

Theorem 5.9 (Moment estimates) Let us assume that m˜ r(γ) < +∞ for some r R and let us set ∈ 1 p +(1 1 )r = p (1 + r/κ) if φ is θ-homogeneous, δ¯ := θ − θ θ (5.13) 1 + r/q p otherwise. ( ≤ ¯ If m˜ δ (σ) < +∞ for some δ δ, and η Mφ,γ [σ], then m˜ δ (η) is ﬁnite and there exists a constant C only depending on≤φ,δ such that∈

m η C m σ m γ Wp σ η ˜ δ ( ) ˜ δ ( ) + ˜ r( ) + φ,γ ( , ) ≤ (5.14)  θ η σ γ 1 1/θ p/ σ η m˜ δ ( ) C m˜ δ ( ) + m˜ r( ) − Wp,α ( , ) .  ≤ Moreover, when δ 1, the topology induced by Wφ γ in Mφ γ [σ] is stronger than the one ≥ , , induced by the Wasserstein distance Wδ . 28

Proof Let us first consider the general case: applying (4.33) we easily obtain (5.14). In order to prove the assertion about the convergence of the moments induced by Wφ,γ (which is equivalent to the convergence in Wδ when δ 1), a simple modification of (4.42) yields ≥ δ δ 1/q p x dη C3 x dσ + m˜ r(γ) Wφ,γ (σ,η) +Wφ γ(σ,η) , (5.15) n n 1 , x 2 | | ≤ x 2 − | | Z| |≥ Z| |≥ which shows that every sequence ηn converging to σ has δ-moments equi-integrable and therefore it is relatively compact with respect to the δ-Wasserstein distance when δ 1. The θ-homogeneous case follows by the same argument and Theorem 4.9. ≥ ⊓⊔ Remark 5.10 There are interesting particular cases covered by the previous result: d 1. When γ(R ) < +∞ then Wφ,γ is always stronger than the 1-Wasserstein distance W1; in θ the -homogeneous case, Wp,α;γ also controls the Wp/θ distance. 2. When mp(γ) < +∞, then Wφ,γ is always stronger than Wp. 3. When φ is θ-homogeneous with θ > 1 and γ is a probability measure with finite moments of arbitrary orders (this is the case of a log-concave probability measure), then all the measures σ Mφ,γ [γ] have finite moments of arbitrary orders and the convergence ∈ d with respect to Wφ,γ yields the convergence in Pδ (R ) for every δ > 0.

5.2 Geometric properties

p Theorem 5.11 (Convexity of the distance and uniqueness of geodesics) Wφ γ ( , ) is con- , · · vex, i.e. for every µ j M+ (Rd),i, j = 0,1, and τ [0,1], if µτ =(1 τ)µ0 + τµ1, i ∈ loc ∈ i − i i p τ τ p 0 0 p 1 1 Wφ γ (µ , µ ) (1 τ)Wφ γ (µ , µ ) + τWφ γ (µ , µ ). (5.16) , 0 1 ≤ − , 0 1 , 0 1

If φ is strictly convex and φ˜ has a sublinear growth w.r.t. ρ (i.e. ϕ˜∞ 0), then for ev- + d ≡ ery µ0, µ1 M (R ) with Wφ γ (µ0, µ1) < +∞ there exists a unique mimimizer (µ,ν) ∈ loc , ∈ CEφ,γ (0,1) of (5.1).

µ j ν j CE µ j µ j τ Proof Let ( , ) φ,γ (0,1; 0 1 ) be two minimizers of (5.1), j = 0,1. For [0,1] µτ τ∈µ0 τµ1 ν τ → τ ν 0 τνν 1 µτ ν τ CE µτ ∈ µτ we set t :=(1 ) t + t , t :=(1 ) t + t . Since ( , ) (0,1; 0 1 ), the convexity of−φ yields − ∈ →

1 1 Wp µτ µτ Φ µτ ν τ γ τ Φ µ0 ν 0 γ τΦ µ1 ν 1 γ φ,γ ( 0 , 1 ) ( t , t )dt (1 ) ( t , t ) + ( t , t ) dt ≤ 0 | ≤ 0 − | | Z Z p 0 0 p 1 1 =(1 τ)Wφ γ (µ , µ ) + τWφ γ (µ , µ ). − , 0 1 , 0 1 φ µτ ρτ γ Let us now suppose that is strictly convex and sublinear. Setting, as usual, t = t + (µτ ) , ν τ = wτ γ, we have for a.e. t (0,1) t ⊥ t t ∈ Φ µτ ν τ γ τ φ ρ0 0 γ τ φ ρ1 1 γ ( t , t ) (1 ) ( t ,wt )d + ( t ,wt )d (5.17) Rd Rd | ≤ − Z Z ρ0 ρ1 0 1 γ Rd µ0 µ1 and the inequality is strict unless t t and wt wt for -a.e. x . If 0 = 0 and µ0 µ1 ≡ ≡ ∈ 1 = 1 , two minimizers should satisfy

ρ0(x) = ρ1(x), w0(x) = w1(x) γ-a.e., ν 0 = ν 1 for L 1-a.e. t (0,1). t t t t t t ∈ 29

Since (µi,ν i) are solutions of the continuity equation, taking the difference we obtain

0 1 0 1 0 1 d ∂t (µ )⊥ (µ )⊥ = ∂t (µ µ ) = ∇ (ν ν ) = 0 in R (0,1). t − t t − t − · t − t × µ0 µ1 The difference ( t )⊥ ( t )⊥ is then independent of time and vanishes at t = 0, so that µ0 = µ1 for every t −[0,1]. t t ∈ ⊓⊔ Theorem 5.12 (Subadditivity) For every µ j M+ (Rd),i, j = 0,1, we have i ∈ loc 0 1 0 1 0 0 1 1 Wφ γ (µ + µ , µ + µ ) Wφ γ (µ , µ ) + Wφ γ (µ , µ ). (5.18) , 0 0 1 1 ≤ , 0 1 , 0 1 In particular

+ d Wφ γ (µ0 + σ, µ1 + σ) Wφ γ (µ0, µ1) σ M (R ). (5.19) , ≤ , ∀ ∈ loc µ j ν j CE µ j µ j Proof Let ( , ) φ,γ (0,1; 0 1 ) be as in the proof of the previous Theorem. Since (µ0 + µ1,ν 0 +ν 1) ∈ CE(0,1; µ0 + µ→1 µ0 + µ1), we get ∈ 0 0 → 1 1 1 1/p W µ0 µ1 µ0 µ1 Φ µ0 µ1 ν 0 ν 1 γ φ,γ ( 0 + 0 , 1 + 1 ) ( t + t , t + t ) dt ≤ 0 | Z 1 1/p 1/p Φ µ0 µ1 ν 0 γ Φ µ0 µ1 ν 1 γ ( t + t , t ) + ( t + t , t ) dt ≤ 0 | | Z 1 h 1/p 1/p i Φ µ0 ν 0 γ Φ µ1 ν 1 γ W µ0 µ0 W µ1 µ1 ( t , t ) + ( t , t ) dt = φ,γ ( 0 , 1 ) + φ,γ ( 0 , 1 ). ≤ 0 | | ⊓⊔ Z h i + d Proposition 5.13 (Rescaling) For every µ0, µ1 M (R ) and λ > 0 we have ∈ loc p λ µ λ µ λ Wp µ µ Wφ,λγ ( 0, 1) = φ,γ ( 0, 1), (5.20)

p λ µ λ µ λ pWp µ µ λ Wφ,γ ( 0, 1) φ,γ ( 0, 1) if 1 p ≤ p ≥ (5.21) Wφ γ (λ µ0,λ µ1) λ Wφ γ (µ0, µ1) if λ 1. ( , ≤ , ≤ Proof (5.20) follows from the corresponding property Φ(λ µ,λνν λγ) = λΦ(µ,ν γ). Anal- ogously, the monotonicity and homogeneity properties of φ yield| |

φ(λρ,λw) φ(ρ,λw) = λ pφ(ρ,w) if λ > 1; ≤ the convexity of φ and the fact that φ(0,0) = 0 yield

φ(λρ,λw) λφ(ρ,w) if λ < 1. ≤ (5.21) follows immediately by the previous inequalities. ⊓⊔ γ γ φ φ µ µ M+ Rd Proposition 5.14 (Monotonicity) If 1 2 and 1 2, then for every 0, 1 loc( ) we have ≥ ≤ ∈ Wφ γ (µ0, µ1) Wφ γ (µ0, µ1). (5.22) 1, 1 ≤ 2, 2 ∞ Rd Theorem 5.15 (Convolution) Let k Cc ( ) be a nonnegative convolution kernel with d ∈ + d Rd k(x)dx = 1 and let kε (x) := ε k(x/ε). For every µ0, µ1 M (R ) we have − ∈ loc R µ µ µ µ ε Wφ,γ kε ( 0 kε , 1 kε ) Wφ,γ ( 0, 1) > 0; (5.23) ∗ ∗ ∗ ≤ ∀ µ µ µ µ limWφ,γ kε ( 0 kε , 1 kε ) = Wφ,γ ( 0, 1). (5.24) ε 0 ∗ ∗ ∗ ↓ 30

Proof Let (µ,ν) CEφ,γ (0,1; µ0 µ1) be an optimal connecting curve as in Theorem 5.11 µε ∈ µ ν ε →ν µε ν ε CE µε µε and let us set t = t kε , t := t kε . Since ( , ) (0,1; 0 1 ), (5.23) then follows by (2.12) whereas∗ (5.24) is a∗ consequence of Theorem∈ 5.6 → ⊓⊔

Remark 5.16 (Smooth approximations) For a given curve (µ,ν) CEφ,γ (0,1) the convolu- ε ε ∈ ν ε tion technique of the previous Theorem exhibits an approximations (µ ,ν ) in CEφ,γ (0,1), ε γ := γ kε , which enjoy some useful properties: ∗ ε ε d ε ε d ε ε ∞ d d 1. µ = ρ L , ν = w L with ρ ,w C (R ); if µ0(R ) < +∞ andm ˜ κ (γ) < +∞ (recall Theorem 5.8), then ρε is also uniformly∈ bounded. − ε ε ε d 2. If supp(k) B1 then ρ ,w are supported in G := x R : dist(x,G) ε , G = supp(γ). ⊂ ∈ ≤ ε ε 3. ρ ,w are classical solution of the continuity equation

ε ε d ∂t ρ + ∇ w = 0 in R (0,1). · × µ ν φ ρε ε γε Φ ρ ν γ Wp µ µ 4. If ( , ) is also a geodesic, ( t ,wt )d ( t, t ) = φ γ ( 0, 1) for every Rd ≤ | , t [0,1]. Z ∈

5.3 Absolutely continuous curves and geodesics

We now study absolutely continuous curves with respect to Wφ,γ and their length. Let us d first recall (see e.g. [3, Chap. 1]) that a curve t µt Mloc(R ), t [0,T], is absolutely 7→ ∈ ∈ continuous w.r.t. Wφ γ if there exists a function m L1(0,T) such that , ∈ t1 W µ µ φ,γ ( t1 , t0 ) m(t)dt 0 t0 < t1 T. (5.25) ≤ Zt0 ∀ ≤ ≤ p The curve has finite p-energy if moreover m L (0,T). The metric derivative µ′ of an absolutely continuous curve is defined as ∈ | | W µ µ µ φ,γ ( t+h, t ) t′ := lim , (5.26) | | h 0 h → | | and it is possible to prove that µ exists and satisfies µ m(t) for L 1-a.e. t (0,T). The | t′| | t′| ≤ ∈ length of µ is then defined as the integral of µ in the interval (0,T). | ′|

Theorem 5.17 (Absolutely continuous curves and their metric velocity) A curve t µt , 7→ t [0,T], is absolutely continuous with respect to Wφ,γ if and only if there exists a Borel ∈ ν d d µ ν family of measures ( t )t (0,T) in Mloc(R ;R ) such that ( , ) CEφ,γ (0,T) and ∈ ∈ T 1/p Φ(µt ,ν t γ) dt < +∞. (5.27) 0 | Z In this case we have

p 1 µ′ Φ(µt ,ν t γ) for L -a.e. t (0,T), (5.28) | t | ≤ | ∈ and there exists a unique Borel family ν˜ t such that

p 1 µ′ = Φ(µt ,ν˜ t γ) for L -a.e. t (0,T). (5.29) | t | | ∈ 31

Proof One implication is trivial: if (µ,ν) CEφ γ (0,T) and (5.27) holds, then (5.5) yields ∈ ,

t1 1/p W µ µ Φ µ ν γ φ,γ ( t1 , t0 ) ( t , t ) dt, (5.30) ≤ t0 | Z so that µ is absolutely continuous and (5.28) holds. Conversely, let us assume that µ is an absolutely continuous curve with length L. A standard reparametrization results [3, Lemma 1.1.4] shows that it is not restrictive to assume N that µ is a Lipschitz map. We ﬁx an integer N > 0, a step size τ := 2− T, and a family of µk,N ν k,N CE τ τ µ µ N geodesics ( , ) φ,γ ((k 1) ,k ; (k 1)τ kτ ), k = 1, ,2 , such that ∈ − − → ···

kτ τΦ µk,N ν k,N γ τ1 p p µ µ µ p ( t , t ) = − Wφ,γ ( (k 1)τ , kτ ) t′ dt. (5.31) | − ≤ (k 1)τ | | Z −

Let (µN,ν N) CEφ γ (0,T) be the curve obtained by gluing together all the geodesics (µk,N,ν k,N). ∈ , Applying Corollary 4.10, we can ﬁnd a subsequence (µNh ,ν Nh ) and a couple (µ,ν) Nh N d d∈ CEφ γ (0,T) such that µ ⇀ µ˜t for every t [0,T] and ν h ⇀ ν in Mloc(R (0,T);R ). , t ∗ ∈ ∗ × It is immediate to check that µt µ˜t for every t [0,T] and ≡ ∈ T T T T µ p Φ µNh ν Nh γ Φ µ ν γ µ p t′ dt liminf ( t , t )dt ( t , t )dt ˜t′ dt, 0 | | ≥ h +∞ 0 | ≥ 0 | ≥ 0 | | Z ↑ Z Z Z which concludes the proof. ⊓⊔ + Corollary 5.18 (Geodesics) For every µ M (Rd) the space Mφ γ [µ] is a geodesic space, ∈ loc , i.e. every couple µ0, µ1 Mφ γ [µ] can be connected by a (minimal, constant speed) geodesic ∈ , t [0,1] µt Mφ γ [µ] such that ∈ 7→ ∈ ,

Wφ γ (µs, µt ) = t s Wφ γ (µ0, µ1) s,t [0,1]. (5.32) , | − | , ∀ ∈ All the (minimal, constant speed) geodesics satisﬁes the continuity equation (4.1) for a Borel ν family of vector valued measures ( t )t (0,1) such that ∈ p Φ(µt,ν t γ) = Wφ γ (µ0, µ1) for a.e. t (0,1). (5.33) | , ∈ If φ is strictly convex and sublinear, geodesics are unique.

Remark 5.19 (A formal differential characterization of geodesics) Arguing as in [27, Chap. µ ρ L d 3], it would not be difﬁcult to show that a geodesic t = t with respect to W2,α;L d should satisfy the system of nonlinear PDE’s in Rd (0,1) × α ∂t ρ + ∇ (ρ ∇ψ) = 0, α · α 1 2 ∂tψ + ρ − ∇ψ = 0,  2 for some potential ψ. Unlike the Wasserstein case, however, the two equations are coupled, and it is not possible to solve the second Hamilton-Jacobi equation in ψ independently of the ﬁrst scalar cosnervation law. In the present paper, we do not explore this direction. 32

We can give a more precise description of the vector measure ν˜ satisfying the optimality condition (5.29). For every measure µ M+ (Rd) we set ∈ loc d d Tanφ,γ (µ) := ν Mloc(R ;R ) : Φ(µ,ν γ) < +∞, ∈ | (5.34) n d d Φ(µ,ν γ) Φ(µ,ν +η γ) η Mloc(R ;R ) : ∇ η = 0 . | ≤ | ∀ ∈ · d d o Observe that for every ν Mloc(R ;R ) such that Φ(µ,ν γ) < +∞ there exists a unique ∈ | d d ν˜ := Π(ν) Tanφ γ (µ) such that ∇ (ν˜ ν) = 0. In fact, the set K(ν) := ν Mloc(R ;R ) : ∈ , · − ′ ∈ ∇(ν ν) = 0 is weakly closed and, by the estimate (3.47) the sublevels of the functional ′ ∗ ν −Φ(µ,ν γ) are weakly relatively compact. Therefore, a minimizer ν˜ exists and it is ′ 7→ ′| ∗ also unique, being Φ(µ, γ) strictly convex. ·| Corollary 5.20 Let (µ,ν) CEφ γ (0,T) so that µ is absolutely continuous w.r.t. Wφ γ . The ∈ , , vector measure ν satisfies the optimality condition (5.29) if and only if ν t Tanφ,γ (µt ) for L 1-a.e. t (0,T). ∈ ∈ Let us consider the particular case of Example 3.5 in the case of a differentiable norm p 2 k · k with associated duality map j1 = D . We denote by jp(w) = w − j1(w) the p-duality map, i.e. the differential of 1 p andk · k we suppose that the concavek k function h : [0,+∞) p k · k → [0,+∞) satisfies 1 limh(r) = lim r− h(r) = 0. (5.35) r 0 r +∞ ↓ ↑ µ M+ Rd γ For every nonnegative Radon measure loc( ) whose support is a subset of supp( ), we define the Radon measure h(µ γ) by ∈ | dµ h(µ γ) := h(ρ) γ where ρ := . (5.36) | · dγ Observe that h(µ γ) γ even if µ is singular w.r.t. γ. | ≪ µ M+ Rd φ Theorem 5.21 Let loc( ) and as in (3.23) with h satisfying (5.35). A vector mea- ν Φ µ ν∈γ ∞ ν µ γ p Rd Rd sure satisfies ( , ) < + iff = vh( ) for some vector field v Lh(µ γ)( ; ). | | ∈ | Moreover, ν Tanφ γ (µ) if and only if the vector field v satisfies ∈ , Lp (Rd;Rd) ∞ d h(µ γ) jp(v) ∇ζ : ζ C (R ) | . (5.37) ∈ ∈ c Proof Being h sublinear, the functional Φ admits the representation

Φ(µ,ν γ) = φ(ρ,w)dγ = h(ρ) w/h(ρ) p dγ = v p dh(µ γ), (5.38) Rd Rd Rd | Z Z k k Z k k | where µ = ργ + µ⊥ and ν = wγ = h(ρ)vγ. The condition ν = vh(µ γ) Tanφ,γ (µ) is then equivalent to | ∈

v p dh(µ γ) v + z p dh(µ γ) z Lp(h(µ γ)) : ∇ zh(µ γ) = 0. Rd k k | ≤ Rd k k | ∀ ∈ | · | Z Z Thanks to the convexity of p, the previous condition is equivalent to ||·|| p d d jp(v) zdh(µ γ) = 0 z L µ γ (R ;R ) : z ∇ζ dh(µ γ) = 0, (5.39) Rd h( ) Rd Z · | ∀ ∈ | Z · | ∇ζ ζ ∞ Rd p Rd Rd i.e. jp(v) belongs to the closure of : Cc ( ) in Lh(µ γ)( ; ). { ∈ } | ⊓⊔ 33

1,p 5.4 Comparison with Wasserstein and W˙ − distances.

d + d Theorem 5.22 If γ(R ) < +∞ then for every µ0, µ1 M (R ) and α < 1 we have ∈ loc d 1/κ W θ (µ0, µ1) W θ γ (µ0, µ1) γ(R ) Wp α;γ (µ0, µ1), (5.40) p/ ≤ p/ ,1; ≤ , where, as usual, θ =(1 α)p + α. − ν Proof Let (µ,ν) CEφ α γ (0,1; µ0 µ1) be an optimal curve, so that ∈ p, , → 1 1 p µ µ Φ µ ν ρ θ p p γ Wp,α;γ ( 0, 1) = p,α ( t , t )dt = ( t ) − wt d dt, (5.41) 0 0 Rd | | Z Z Z where µt := ρt γ + µ , ν t = wt γ γ. H¨older inequality yields t⊥ ≪ 1 p/θ µ µ ρ 1 p/θ p/θ γ γ Rd 1 1/θ p/θ µ µ Wp/θ,1;γ ( 0, 1) ( t ) − wt d dt ( ) − Wp,α;γ ( 0, 1). ≤ 0 Rd | | ≤ ⊓⊔ Z Z

Theorem 5.23 Let us suppose that m˜ k(γ) < +∞, κ = p/(θ 1) = q/(1 α), let µ0, µ1 P(Rd), and let κ = κ/(κ 1) be the− Holder’s¨ conjugate exponent− of κ. Then− ∈ ∗ −

µ0 µ1 1,κ = Wκ ,0;γ (µ0, µ1) Wp,α;γ (µ0, µ1). (5.42) k − kW˙ γ− ∗ ∗ ≤ Proof We keep the same notation of the previous Theorem, setting τ θ 1 1 p τ = p/r := 1 + p θ, τ∗ := = 1 + , x =(τ∗)− = − . − τ 1 p θ 1 + p θ − − − d Observing that µt P(R ) thanks to Theorem 4.9, we obtain ∈ 1 1 r µ µ r γ ρ x ρ x r γ Wr,0;γ ( 0, 1) wt d dt = t t − wt d dt ≤ 0 Rd | | 0 Rd | | Z Z Z Z 1 1/τ 1 1/τ ρ xτ rτ γ ρθ p p γ r µ µ − wt d dt = − wt d dt = Wp,α;γ ( 0, 1). ≤ 0 Rd | | 0 Rd | | ⊓⊔ Z Z Z Z γ M+ Rd Theorem 5.24 (Comparison with Wp) Assume that loc( ) is a bounded pertur- V d ∈ bation of a log-concave measure (e.g. γ = f e− L , where V is a convex function and f d ∞ nonnegative and bounded). If µi = siγ P(R ) with si L (γ) and mp(µi) L < +∞ then ∈ ∈ ≤ Wφ,γ (µ0, µ1) < +∞ and there exists a constant C only depending on L, φ, and γ such that

Wφ γ (µ0, µ1) CWp(µ0, µ1). (5.43) , ≤ Proof It is not restrictive to assume that γ is log-concave. We can then consider the optimal + d d plan Σ M (R R ) induced by the p-Wasserstein distance (1.1) between µ0 and µ1 and ∈ × the interpolant µt deﬁned as

d d d µt (A) = Σ (x0,x1) R R : (1 t)x0 +tx1 A A B(R ). (5.44) { ∈ × − ∈ } ∀ ∈

It is possible to prove (see e.g. [3, Theorems 7.2.2, 8.3.1, 9.4.12]) that µt is the geodesic interpolant between µ0 and µ1, it satisﬁes the continuity equation

d ∂t µt + ∇ ν t = 0 in R (0,1) · × 34

with respect to a vector valued measure ν t = vt µt µt where the vector ﬁeld vt satisﬁes ≪ 1 1 Φ µ ν p µ p µ µ p( t , t )dt = vt(x) d t (x)dt = Wp ( 0, 1), 0 0 Rd | | Z Z Z

µ γ ∞ and ﬁnally t = st with st L (γ) L := max s0 L∞(γ), s1 L∞(γ) . Observe that, being dk k ≤ k k k k st (x) L for γ-a.e. x R and φ(0,0) = 0, Theorem 3.1 yields ≤ ∈ st p φ(st,st vt ) φ(L,Lvt ) CLst vt γ-a.e., ≤ L ≤ | | so that

p p Φ(µt ,ν t ) = φ(st,stvt )dγ(x) CL vt st dγ = CL vt dµt , Rd Rd Rd Z ≤ Z | | Z | | and therefore

1 1 Wp µ µ Φ µ ν C p µ C p µ µ φ,γ ( 0, 1) ( t, t )dt L vt d t dt = LWp ( 0, 1). ≤ 0 ≤ 0 Rd | | ⊓⊔ Z Z Z d d ∞ Corollary 5.25 If µi = siL P(R ) have L -densities with compact support (or, more ∈ W µ µ ∞ generally, ﬁnite p-momentum), then φ,L d ( 0, 1) < + .

d Theorem 5.26 If µi = siγ with si L > 0 γ-a.e. in R , then there exists a constant C depending on L and φ such that ≥

Wφ,γ (µ0, µ1) CL µ0 µ1 1,p . (5.45) ≤ k − kW˙ γ−

p d d Proof Let us ﬁrst observe that if µ0 µ1 1,p < +∞ then there exists w Lγ (R ;R ) k − kW˙γ− ∈ such that ∇ ν µ µ ν γ p γ µ µ p = 1 0, := w , w d = 0 1 1,p . (5.46) Rd W˙ γ− − · − Z | | k − k q d d In fact, in the Banach space X := Lγ (R ;R ) we can consider the linear space Y := Dζ : ζ C1(Rd) and the linear functional ∈ c

ζ µ µ ζ ζ 1 Rd ℓ,y := d( 1 0) if y = D for some Cc ( ). Rd h i Z − ∈

ℓ is well deﬁned and satisﬁes ℓ,y µ0 µ1 1,p y q Rd Rd for every y Y. Hahn- h i ≤ k − kW˙γ− k kLγ ( ; ) ∈ Banach Theorem and Riesz representation Theorem yield the existence of w Lp(Rd;Rd) ∈ such that ℓ,y = Rd w ydγ, which yields (5.46). Setting µt =(1 t)µ0 + tµ1, it is then h i · − immediate to check that (µt ,ν ) CE(0,1; µ0 µ1); we can then compute R ∈ → 1 Wp µ µ φ γ φ γ p γ φ,γ ( 0, 1) ((1 t)s0 +ts1,w)d dt (L,w)d CL w d , ≤ 0 Rd − ≤ Rd ≤ Rd | | Z Z Z Z

where we used the fact that (1 t)s0 + ts1 L γ-almost everywhere and the map ρ − ≥ 7→ φ(ρ,w) is nonincreasing. ⊓⊔ 35

5.5 The case γ = L d and the Heat equation as gradient ﬂow

One of the most interesting cases corresponds to the choice

d α α α p γ := L , hα (ρ) := ρ , 0 < α < 1, φp α (ρ,w) := ρ w/ρ . (5.47) , | | In this case the expression of the weighted Wasserstein distance becomes

1 p µ µ ρα p ∂ µ ∇ ρα Rd W α L d ( 0, 1) := min t vt dxdt : t + ( v) = 0 in (0,1) p, ; 0 Rd | | · × nZ Z µ = ρ L d + µ , µ = µ , µ = µ . t t t⊥ t 0 0 t 1 1 | = | = o P Rd κ p q The metric Wp,α;L d restricted to ( ) is complete if d < = θ 1 = 1 α . − − P Rd κ Remark 5.27 ( ( ) is not complete w.r.t. Wp,α;L d if d > ) The above condition is almost sharp; here is a simple counterexample in the case d > κ. We consider an initial probability d ∞ d measure with compact support µ0 = ρ0L , ρ0 L (R ), and, for t 0, the family ∈ ≥ d dt t d µt := ρt L , ρt (x) := e− ρ0(e− x), ν t := xµt = xρt (x)L . (5.48)

d It is easy to check that (µ,ν) CE(0,+∞), µt (R ) = 1. Evaluating the functional Φt := d ∈ Φp α (µt ,ν t L ) we get , | Φ ρθ p ρ p dθt ρθ dt p t = t − t x dx = e− 0 (e− x) x dx Rd | | Rd | | Z Z dt dθt+pt dt ρθ dt dt p (d(1 θ)+p)t ρθ p = e − e− 0 (e− x) e− x dx = e − 0 (y) y dy Rd | | Rd | | Z Z so that +∞ κ Φ1/p (1 d/κ)t Φ1/p ∞ κ t = ce − , t dt = c < + if d > . 0 d κ Z − d If d > κ we obtain a curve t µt P(R ) of ﬁnite length w.r.t. W α L d (in particular 7→ ∈ p, ; (µn)n N is a Cauchy sequence) such that lim µt = 0 in the weak∗ topology. ∈ t +∞ ↑ In the remaining part of this section, we want to study the properties of Wp,α;L d with respect to the heat ﬂow. We thus introduce

1 x 2/4 1 x 2/4t d/2 g(x) = g1(x) = e−| | , gt (x) := e−| | = t− g1(x/√t), (4π)d/2 (4πt)d/2

+ d and for every µ M (R ) withm ˜ δ (µ) < +∞ for some δ 0, we set ∈ loc ≤ d St [µ] = µ gt = ut L , ut (x) = St [µ](x) := gt (x y)dµ(y). (5.49) ∗ Rd − Z It is well known that u C∞(Rd (0,+∞)) and ∈ × d ∂t u ∆u = 0 in R (0,+∞), St [µ]⇀∗µ as t 0. (5.50) − × ↓ µ0 µ1 M+ Rd m µi ∞ Theorem 5.28 (Contraction property) Let , loc( ) with ˜ δ ( ) < + and W µ1 µ2 ∞ µi S µi ∈ φ,L d ( , ) < + . If t := t [ ] are the corresponding solutions of the heat ﬂow, then

1 2 1 2 Wφ L d (µ , µ ) Wφ L d (µ , µ ) t > 0. (5.51) , t t ≤ , ∀ 36

Proof It sufﬁcient to approximate the Gaussian kernel g by a family of C∞ kernels kn with compact support and then apply Theorem 5.15, observing that kn L d = L d. ∗ ⊓⊔ α We consider now the particular case of the W2,α;L d weighted distance with > 1 2/d. Let us ﬁrst introduce the convex density function (recall (1.9)) −

1 2 α 1 α ψα (ρ) := ρ − , such that ψα′′(ρ) = = ρ− , (5.52) (2 α)(1 α) h(ρ) − − and the corresponding entropy functional

d d d Ψα (µ) = Ψα (µ L ) := ψα (ρ)dx, if µ = ρL L . (5.53) Rd | Z ≪ We also introduce the set Q := µ P(Rd) : Ψ(µ) < +∞ . ∈ d d Theorem 5.29 If µ P(R ) then µt = St [µ] = ut L Q for every t > 0, the map t ∈ ∈ 7→ Ψα (µt ) is nonincreasing, and it satisﬁes the energy identity

t Ψα (µt ) + Φ2,α (ur,∇ur)dr = Ψα (µs) 0 < s t < +∞; (5.54) s ∀ ≤ Z when µ Q then the previous identity holds even for s = 0. Moreover, µt satisﬁes the Evo- lution Variational∈ Inequality

+ 1 d 2 µ σ Ψ µ Ψ σ σ W d ( t , ) + α( t ) α ( ) t 0, Q. (5.55) 2 dt 2,α;L ≤ ∀ ≥ ∀ ∈ α Proof Since ψα′′(u) = u− , a direct computation shows

d ψ d ∇ ∇ψ ∇ 2 α Φ ∇ α (ut )dx = ut α′ (ut )dx = ut ut− dx = 2,α (ut , ut ). dt Rd dt Rd Rd Z − Z · Z | | Concerning (5.55), we use the technique introduced by [12, § 2]: we consider a geodesic σ ν CE σ µ σ Rd ( s, s)s [0,1] (0,1; ), which satisﬁes s( ) = 1 by Theorem 4.9. We set ∈ ∈ → ε ε d ε ε d ε ε ε σ = u L := Sε st[σs], ν˜ = w˜ L := Sε st[ν s], w := w˜ t∇u . s,t s,t + s,t s,t + s,t s,t − s,t It is not difﬁcult to check that

ε ε d ∂su + ∇ w = 0 in R (0,1), (5.56) s,t · s,t × so that

1 α 2 µ σ ε ε ε ε 2 Φ σ ε ν ε L d W α L d ( ε+t , ) As,t ds, As,t := us,t − ws,t dx = 2,α ( s,t, s,t ). 2, ; Rd ≤ Z0 Z | | | We thus evaluate

ε ε α ∇ ε ε ε 2 2 ∇ ε 2 As,t = us,t − 2t us,t ws,t + w˜ s,t t us,t dx Rd − · | | − | | Z ε α ∇ ε ε ε α ε 2 2t us,t − us,t ws,t dx + us,t − w˜ s,t dx Rd Rd ≤ − Z · Z | | ε d 2t ∂sΨ(σ ) + Φ2 α (σs,ν s L ), (5.57) ≤ − s,t , | 37

where we used the facts

∂ ψ ε (5.56) ∇ψ ε ε (5.52) ε α ∇ ε ε s α (us,t )dx = α′ (us,t ) ws,t dx = us,t − us,t ws,t dx, Rd Rd · Rd · Z Z Z α ε ε 2 Φ σ ν L d Φ σ ν L d us,t − w˜ s,t dx = 2,α ( s gε+st , s gε+st ) 2,α ( s, s ) Rd | | ∗ ∗ | ≤ | Z thanks to the convolution contraction property of Theorem 2.3. Integrating (5.57) with σ ν respect to s from 0 to 1 and recalling that ( s, s)s [0,1] is a minimal geodesic and that σ ε µ σ ε σ ∈ 1,t = ε+t and 0,t = , we get

1 ε 2 Ψα µε Ψα σ µ σ As,t ds + 2t ( +t ) 2t ( ) +W2,α;L d ( , ). (5.58) 0 ≤ Z We deduce that

1 2 1 2 W α (µε t ,σ) +tΨ(µε t ) tΨ(σ) + W α (µ,σ). (5.59) 2 2, + + ≤ 2 2, Passing to the limit as ε 0 and then as t 0 after dividing the inequality by t we get (5.55) ↓ ↓ at t = 0. Recalling the semigroup property of the heat equation, we obtain (5.55) for every time t 0. ≥ ⊓⊔ (5.55) is the metric formulation of the gradient ﬂow of the (geodesically convex) functional Ψ Q α in the metric space ( ,W2,α;L d ), see [3, Chap. 4]. Applying [12, Theorem 3.2] we eventually obtain:

d d Corollary 5.30 (Geodesic convexity of Ψα ) Let α > 1 2/d, µi = ρiL P(R ) with µ µ ∞ ρ2 α ∞ µ− ∈ W2,α;L d ( 0, 1) < + and Rd i − dx < + , and let t ,t [0,1], be the minimal speed ∈ d d geodesic connecting µ0 to µ1 w.r.t. W α L d . Then for every t [0,1] µt = ρt L L R 2, ; ∈ ≪ ρ2 α ρ2 α ρ2 α t − dx (1 t) − dx +t − dx. (5.60) Rd Rd 0 Rd 1 Z ≤ − Z Z Acknowledgements. J.D. and B.N. have been partially supported by the ANR project IFO. B.N. has also been partially supported by the ANR project OTARIE.

References

1. L.AMBROSIO AND G. BUTTAZZO, Weak lower semicontinuous envelope of functionals defined on a space of measures, Ann. Mat. Pura Appl. (4), 150 (1988), pp. 311–339. 2. L.AMBROSIO,N. FUSCO, AND D. PALLARA, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. 3. L.AMBROSIO,N.GIGLI, AND G. SAVARE´, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. 4. L. AMBROSIO AND F. SANTAMBROGIO, Necessary optimality conditions for geodesics in weighted Wasserstein spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), pp. 23–37. 5. L.AMBROSIO,G. SAVARE´ , AND L.ZAMBOTTI, Existence and stability for Fokker-Planck equations with log-concave reference measure, ArXiv Mathematics e-prints, (2007). 6. A.ARNOLD, P. MARKOWICH,G.TOSCANI, AND A. UNTERREITER, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), pp. 43–100. 7. J.-D. BENAMOU AND Y. BRENIER, A computational fluid mechanics solution to the Monge- Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375–393. 38

8. A. BRANCOLINI,G.BUTTAZZO, AND F. SANTAMBROGIO, Path functionals over Wasserstein spaces, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415–434. 9. Y. BRENIER, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), pp. 375–417. 10. G. BUTTAZZO,C. JIMENEZ, AND E. OUDET, An optimization problem for mass transportation with congested dynamics, Preprint, available online at http://cvgmt.sns.it, (2007). 11. J. CARRILLO,S.LISINI, AND G. SAVARE´, in preparation, (2008). 12. S. DANERI AND G. SAVARE´, Eulerian calculus for the displacement convexity in the Wasserstein distance, Preprint, arXiv:0801.2455v1, ((2008)). 13. E. DE GIORGI, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, 1993, pp. 81–98. 14. C. DELLACHERIE AND P.-A. MEYER, Probabilities and potential, vol. 29 of North-Holland Mathemat- ics Studies, North-Holland Publishing Co., Amsterdam, 1978. 15. J. DOLBEAULT,B.NAZARET, AND G. SAVARE´, in preparation, (2008). 16. L.C.EVANS, Partial differential equations and Monge-Kantorovich mass transfer, in Current develop- ments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 65–126. 17. L.C.EVANS AND W. GANGBO, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), pp. viii+66. 18. W. GANGBO AND R. J. MCCANN, The geometry of optimal transportation, Acta Math., 177 (1996), pp. 113–161. 19. R. JORDAN, D. KINDERLEHRER, AND F. OTTO, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), pp. 1–17 (electronic). 20. P. MARECHAL´ , On a functional operation generating convex functions. I. Duality, J. Optim. Theory Appl., 126 (2005), pp. 175–189. 21. , On a functional operation generating convex functions. II. Algebraic properties, J. Optim. The- ory Appl., 126 (2005), pp. 357–366. 22. J. NASH, C1 isometric imbeddings., Ann. of Math., (1954). 23. , The imbedding problem for Riemannian manifolds., Ann. of Math., (1956). 24. F. OTTO, Doubly degenerate diffusion equations as steepest descent, Manuscript, (1996). 25. , Evolution of microstructure in unstable porous media ﬂow: a relaxational approach, Comm. Pure Appl. Math., 52 (1999), pp. 873–915. 26. , The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), pp. 101–174. 27. F. OTTO AND C. VILLANI, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), pp. 361–400. 28. S. T. RACHEV AND L. RUSCHENDORF¨ , Mass transportation problems. Vol. I, Probability and its Ap- plications, Springer-Verlag, New York, 1998. Theory. 29. R. ROCCKAFELLAR, A general correspondence between dual minimax problems and convex problems, Paciﬁc J. of Math., 25 (1968), pp. 597–611. 30. C. VILLANI, Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003. 31. C. VILLANI, Optimal Transport, Old and New, Springer Verlag, To appear.